# Enumerating Galois extensions of number fields

Robert J. Lemke Oliver

###### Abstract.

Let $k$ be a number field. We provide an asymptotic formula for the number of
Galois extensions of $k$ with absolute discriminant bounded by some $X\geq 1$,
as $X\to\infty$. We also provide an asymptotic formula for the closely related
count of extensions $K/k$ whose normal closure has discriminant bounded by
$X$. The key behind these results is a new upper bound on the number of Galois
extensions of $k$ with a given Galois group $G$ and discriminant bounded by
$X$; we show the number of such extensions is
$O_{[k:\mathbb{Q}],G}(X^{\frac{4}{\sqrt{|G|}}})$. This improves over the
previous best bound $O_{k,G,\epsilon}(X^{\frac{3}{8}+\epsilon})$ due to
Ellenberg and Venkatesh. In particular, ours is the first bound for general
$G$ with an exponent that decays as $|G|\to\infty$.

##  1\. Introduction

For any $X\geq 1$, let $\mathcal{F}(X):=\\{K/\mathbb{Q}:|\mathrm{Disc}(K)|\leq
X\\}$ be the set of number fields of any degree whose absolute discriminant is
at most $X$. It is generally expected that there should be some constant $c>0$
so that $\\#\mathcal{F}(X)\sim cX$ as $X\to\infty$, but this appears to be far
out of reach; to date, the best known upper bound is $\\#\mathcal{F}(X)\leq
2X^{9(\log\log X)^{3}}$ [Lem23, Corollary 1.9], which is not even polynomial
in $X$. The primary challenge faced in obtaining a polynomial bound is the
consideration of fields of arbitrarily large degree.

Here, we overcome this difficulty in a natural setting by establishing an
asymptotic for the number of Galois extensions of $\mathbb{Q}$, or, in fact,
of any number field, and by counting all number fields when ordered by the
discriminant of their normal closure.

More concretely, let $k$ be a number field, and for any $X\geq 1$, let
$\mathcal{F}_{k}^{\mathrm{Gal}}(X):=\\{K/k\text{
Galois}:|\mathrm{Disc}(K)|\leq X\\}.$

######  Theorem 1.1.

For any number field $k$, any $X\geq 1$, and any $\epsilon>0$, we have

(1.1)
$\\#\mathcal{F}_{k}^{\mathrm{Gal}}(X)=\frac{\mathrm{Res}_{s=1}\zeta_{k}(s)}{2^{r_{2}(k)}\zeta_{k}(2)|\mathrm{Disc}(k)|^{2}}X+O_{k,\epsilon}(X^{1-\delta_{k}+\epsilon})$

where $\zeta_{k}(s)$ is the Dedekind zeta function for $k$ and we have set
$\delta_{k}=\frac{1}{2}$ if $[k:\mathbb{Q}]\leq 3$ and
$\delta_{k}=\frac{2}{[k:\mathbb{Q}]+1}$ in general. If $k=\mathbb{Q}$, then we
moreover have

(1.2)
$\\#\mathcal{F}_{\mathbb{Q}}^{\mathrm{Gal}}(X)=\frac{6}{\pi^{2}}X+P_{2}(\log
X)\cdot X^{1/2}+O\left(X^{1/2}\exp\left(-c\cdot(\log X)^{3/5}(\log\log
X)^{-1/5}\right)\right)$

where $P_{2}$ is an explicitly computable polynomial of degree $2$ and where
$c>0$ is an absolute constant.

We also prove that the same asymptotic formula holds for the closely related
set
$\mathcal{F}_{k}^{\mathrm{nc}}(X):=\\{K/k:|\mathrm{Disc}(\widetilde{K})|\leq
X\\},$ where $\widetilde{K}$ is the normal closure of $K/k$, and where we view
extensions as living inside a fixed choice of algebraic closure
$\overline{k}$.

######  Theorem 1.2.

For any number field $k$, any $X\geq 1$, and any $\epsilon>0$, we have

(1.3)
$\\#\mathcal{F}_{k}^{\mathrm{nc}}(X)=\frac{\mathrm{Res}_{s=1}\zeta_{k}(s)}{2^{r_{2}(k)}\zeta_{k}(2)|\mathrm{Disc}(k)|^{2}}X+O_{k,\epsilon}(X^{1-\delta_{k}+\epsilon})$

where $\zeta_{k}(s)$ is the Dedekind zeta function for $k$ and we have set
$\delta_{k}=\frac{1}{2}$ if $[k:\mathbb{Q}]\leq 3$ and
$\delta_{k}=\frac{2}{[k:\mathbb{Q}]+1}$ in general. If $k=\mathbb{Q}$, then we
moreover have

(1.4)
$\\#\mathcal{F}_{\mathbb{Q}}^{\mathrm{nc}}(X)=\frac{6}{\pi^{2}}X+P_{2}(\log
X)\cdot X^{1/2}+O\left(X^{1/2}\exp\left(-c\cdot(\log X)^{3/5}(\log\log
X)^{-1/5}\right)\right)$

where $P_{2}$ and $c>0$ are the polynomial of degree $2$ and absolute constant
appearing in Theorem 1.1, respectively.

The asymptotic for $\mathcal{F}_{k}^{\mathrm{Gal}}(X)$ is the first result
counting a subset of $\mathcal{F}(X)$ containing fields of arbitrarily large
degree. Moreover, we might argue that
$\mathcal{F}_{\mathbb{Q}}^{\mathrm{Gal}}(X)$ is the most natural proper subset
of $\mathcal{F}(X)$. Additionally, the asymptotic for
$\mathcal{F}_{\mathbb{Q}}^{\mathrm{nc}}(X)$ is the first result counting _all_
number fields, irrespective of degree, by _some_ natural invariant (in this
case, the discriminant of the normal closure).

######  Remark.

It is often common in the field counting literature to weigh fields inversely
to the size of their automorphism group. For our purposes, it is a strictly
harder problem to consider the unweighted version, with a weighted version
following straightforwardly. Similarly, one may wish to consider fields in
$\mathcal{F}_{k}^{\mathrm{nc}}(X)$ only up to isomorphism and not inside a
fixed algebraic closure $\overline{k}$. This too is a strictly easier problem,
as are those problems arising from placing restrictions on the Galois groups
(e.g., that they be abelian, nilpotent, or solvable). We comment further on
these variations in §5.3.

The main terms in Theorem 1.1 and Theorem 1.2 will be familiar to experts, as
they arise simply from the count of quadratic extensions $K/k$, which was
first established over a general number field in [DW88, Theorem 4.2]; in fact,
the error terms in (1.1) and (1.3) arise from the best known error term in
this count [MT23, Theorem 2]. Similarly, the term $P_{2}(\log X)\cdot X^{1/2}$
in (1.2) and (1.4) accounts for the known asymptotic number of $C_{3}$,
$C_{4}$, and $C_{2}\times C_{2}$ extensions of $\mathbb{Q}$ [Coh54, Bai80,
Mäk85, Wri89, FLN18]. More generally, given a finite group $G$, we may define

$\mathcal{F}_{k}(X;G):=\\{K/k\text{ Galois}:\mathrm{Gal}(K/k)\simeq
G,|\mathrm{Disc}(K)|\leq X\\},$

and we observe that

$\mathcal{F}_{k}^{\mathrm{Gal}}(X)=\bigcup_{G}\mathcal{F}_{k}(X;G).$

A similar expression holds also for $\\#\mathcal{F}_{k}^{\mathrm{nc}}(X)$, but
where the contribution of each group $G$ is weighted by the number of core-
free subgroups $H\leq G$. (Recall that a subgroup is _core-free_ if the
intersection of its conjugates is trivial. Every group has at least one core-
free subgroup, namely the trivial group.)

A conjecture of Malle [Mal04] predicts that
$\\#\mathcal{F}_{k}(X;G)\ll_{k,G,\epsilon}X^{\frac{p}{(p-1)|G|}+\epsilon}$,
where $p$ is the least prime dividing $|G|$. If this conjecture is true, then
one should expect the contribution from groups $G$ with order at least $5$ to
land in the error terms of Theorems 1.1 and 1.2 – but only provided this bound
holds sufficiently uniformly in $G$, as a priori the Minkowksi bound implies
that one must consider groups $G$ of order up to a constant times $\log X$.
This form of Malle’s conjecture is known in $X$ if $G$ is abelian or nilpotent
[Mäk85, Wri89, KM04], but remains out of reach for general $G$. The previous
best general bound in the literature on $\\#\mathcal{F}_{k}(X;G)$ is in work
of Ellenberg and Venkatesh [EV06, Proposition 1.3], and is of the form
$\\#\mathcal{F}_{k}(X;G)\ll_{k,G,\epsilon}X^{\frac{3}{8}+\epsilon}$ for every
$G$ with order at least $5$. This bound provides a sufficient bound in terms
of $X$, but its dependence on the group $G$ is not explicated.

Thus, the main technical ingredient leading to Theorems 1.1 and 1.2 is the
following fully explicit bound on $\\#\mathcal{F}_{k}(X;G)$, that additionally
incorporates a substantial asymptotic improvement, and whose dependence on the
group $G$ is sufficient for our purposes.

######  Theorem 1.3.

There is a positive absolute constant $c_{1}$ such that for any finite group
$G$ and any number field $k$, there holds for every $X\geq 1$

$\\#\mathcal{F}_{k}(X;G)\leq e^{d|G|}\cdot(2d|G|^{2})^{c_{1}d|G|^{1/2}}\cdot
X^{\frac{6}{\sqrt{|G|}}}$

where $d=[k:\mathbb{Q}]$. Explicitly, we may take $c_{1}=18.5$.

There is always a balance to be found between aesthetics and sharpness in
stating an explicit result. For Theorem 1.3 specifically, we have prioritized
separating the dependence on $X$ and the parameters $G$ and $k$, and we have
not tried to optimize the admissible value of $c_{1}$. Our next result
provides a bound with a smaller power of $X$, at the expense of making the
dependence of the implied constant on the group $G$ and the degree of $k$
inexplicit.

######  Theorem 1.4.

For any finite group $G$, any number field $k$, and any $X\geq 1$, there holds

$\\#\mathcal{F}_{k}(X;G)\ll_{[k:\mathbb{Q}],G}X^{\frac{4}{\sqrt{|G|}}}.$

In fact, there holds
$\\#\mathcal{F}_{k}(X;G)\ll_{[k:\mathbb{Q}],G}X^{\frac{c}{\sqrt{|G|}}}$ with
$c=\frac{6935}{18\sqrt{9690}}=3.913\dots$.

Finally, we provide an even stronger bound in terms of $X$, but where the
dependence of the implied constant on the base field $k$ is fully inexplicit
and is allowed to depend on the discriminant of $k$, for example.

######  Theorem 1.5.

Let $c_{0}=\frac{863441}{2880\sqrt{9690}}\approx 3.045$. For any finite group
$G$, any number field $k$, any $X\geq 1$, and any $\epsilon>0$, there holds

$\\#\mathcal{F}_{k}(X;G)\ll_{k,G,\epsilon}X^{\frac{c_{0}}{\sqrt{|G|}}+\epsilon}.$

######  Remark.

As suggested by their somewhat ad hoc expressions, the constants $c$ and
$c_{0}$ in Theorems 1.4 and 1.5 arise from analyzing the groups $G$ for which
our methods are weakest; the specific constants $c$ and $c_{0}$ in the
statements arise from the sporadic simple group $\mathrm{J}_{3}$. By imposing
constraints on the composition factors of the groups $G$ considered (e.g.,
that there is no composition factor isomorphic to $\mathrm{J}_{3}$), it it
possible to reduce the value of the constants $c$ and $c_{0}$, but only
slightly. In particular, it will be necessary to treat groups $G$ with an
abelian socle differently than we do below to provide a constant $c_{0}$ less
than $2$.

Moreover, even for groups as simple as
$\mathbb{F}_{p}\rtimes\mathbb{F}_{p}^{\times}$, at present we do not know how
to provide an exponent that is $O(|G|^{-\frac{1}{2}-\delta})$ for some
$\delta>0$ independent of $p$. Indeed, such a result would require substantial
progress toward the so-called $\ell$-torsion conjecture on class groups, and
at a level that is well beyond the scope of existing methods. Thus,
improvements to the qualitative shape of Theorem 1.5 will require at least a
breakthrough in our understanding of class groups of number fields. In this
sense, despite the fact that the stated version of Theorem 1.5 is technically
limited by our understanding of the simple group $\mathrm{J}_{3}$, the shape
of the theorem is more fundamentally limited by our understanding of groups
with abelian socles and the closely related problem of bounding torsion
subgroups of class groups. See Theorem 4.20 below.

One natural application where Theorem 1.5 is of use is to bounding the number
of “Galois” polynomials.

######  Corollary 1.6.

For any $n\geq 5$ and any $H\geq 1$, the number of monic, irreducible, degree
$n$ polynomials $f\in\mathbb{Z}[X]$ with coefficients bounded by $H$ in
absolute value such that $|\mathrm{Gal}(f)|=n$, is
$O_{n,\epsilon}(H^{2c_{0}\sqrt{n}+1-\frac{2c_{0}}{\sqrt{n}}+\epsilon})$, where
$c_{0}$ is as in Theorem 1.5.

###### Proof.

This follows immediately from Theorem 1.5 and [LOT20, Theorem 1.3], though see
also [Bha24] for more on the history of this problem. ∎

Theorems 1.3–1.5 (and also [EV06, Proposition 1.3] before them, though less
substantially) make use of the classification of finite simple groups. Without
using the classification, we show it is at least possible to obtain an
exponent of $X$ less than $1$.

######  Theorem 1.7.

Let $G$ be a finite group of order at least $3$. Then for any number field $k$
and any $X\geq 1$, there holds for every $\epsilon>0$

$\\#\mathcal{F}_{k}(X;G)\ll_{[k:\mathbb{Q}],G,\epsilon}X^{1-\frac{1}{4|G|}+\epsilon}.$

This does not rely on the classification of finite simple groups.

Theorem 1.7 essentially follows from the methods of [Lem23], as we explain in
§6, and treats most groups $G$ simultaneously. For groups of order divisible
by either $2$ or $3$, it is sometimes necessary to consider a particular
quotient of $G$. In carrying this out for groups of odd order divisible by
$3$, we invoke the Feit–Thompson theorem that groups of odd order are
solvable. The Feit–Thompson theorem is still a rather heavy hammer, so we
leave open the question of whether there is a more elementary proof of a bound
with an exponent strictly less than $1$ for every group $G$.

In contrast to the proof of Theorem 1.7, the proof of Theorem 1.3 proceeds by
induction, exploiting the minimal normal subgroups of groups $G$. It is
naturally in our treatment of nonabelian minimal normal subgroups that we
appeal to the classification of finite simple groups. Beyond that, though, the
proofs of Theorems 1.1 and 1.2 also rely on a result of Holt [Hol89] that
provides an upper bound on the number of finite groups of bounded order.
Holt’s proof also relies on the classification, and while a weaker result
would still afford a proof of our first theorems, we nevertheless expect any
proof of Theorems 1.1 and 1.2 to necessarily pass through the classification.

## Acknowledgements

The author would like to thank Pham Huu Tiep for many useful conversations on
earlier versions of the underlying ideas. He would also like to thank Arul
Shankar, Frank Thorne, and Jesse Thorner for useful comments on a preliminary
version of this paper.

The author was supported by a grant from the National Science Foundation
(DMS-2200760) and by a Simons Foundation Fellowship in Mathematics.

##  2\. Preliminary reductions and an inductive strategy

As indicated above, we approach the proof of Theorems 1.3–1.5 by induction. In
this section, we lay out this strategy more precisely, and we indicate how the
general problem can be substantially reduced to considering only certain
classes of groups $G$. In particular, unlike previous inductive approaches to
related problems that essentially rely on a single normal subgroup, including
[KM04, Klü12, Wan21, Alb20], our focus in this section is on groups with at
least two minimal normal subgroups. We will ultimately show that given such a
group $G$, the study of $G$-extensions can be nearly reduced to the study of
various $G_{i}$-extensions, where $G_{i}$ is some quotient of $G$ with a
unique minimal normal subgroup. The mechanics realizing this reduction are
almost trivial, but at least for the purposes of this paper, its impacts are
far from it.

The key to our approach is the following lemma.

######  Lemma 2.1.

Let $G$ be a finite group, and suppose that $G$ admits two normal subgroups
$N_{1},N_{2}\trianglelefteq G$ such that $N_{1}\cap N_{2}$ is trivial. Then
for any number field $k$ and any $X\geq 1$, we have

$\\#\mathcal{F}_{k}(X;G)\leq\\#\mathcal{F}_{k}(X^{\frac{1}{|N_{1}|}};G/N_{1})\cdot\\#\mathcal{F}_{k}(X^{\frac{1}{|N_{2}|}};G/N_{2}).$

###### Proof.

Suppose that $K\in\mathcal{F}_{k}(X;G)$. Let $K^{N_{1}}$ and $K^{N_{2}}$
denote the subfields of $K$ fixed by $N_{1}$ and $N_{2}$, respectively. By the
assumption that $N_{1}\cap N_{2}$ is trivial, we find that the compositum of
$K^{N_{1}}$ and $K^{N_{2}}$ must be $K$, and hence that $K$ is determined by
these two subfields. Moreover, since $[K:K^{N_{1}}]=|N_{1}|$, we have that
$|\mathrm{Disc}(K)|\geq|\mathrm{Disc}(K^{N_{1}})|^{|N_{1}|}$, and hence
$|\mathrm{Disc}(K^{N_{1}})|\leq X^{\frac{1}{|N_{1}|}}$. Similarly, we have
$|\mathrm{Disc}(K^{N_{2}})|\leq X^{\frac{1}{|N_{2}|}}$, and the result
follows. ∎

In particular, any group $G$ admitting two “large” normal subgroups satisfying
the hypothesis of Lemma 2.1 may be readily handled by induction.

######  Lemma 2.2.

Let $G$ be a finite group, and suppose that $N_{1},N_{2}\trianglelefteq G$ are
normal subgroups with $N_{1}\cap N_{2}=1$ and
$\frac{1}{\sqrt{|N_{1}|}}+\frac{1}{\sqrt{|N_{2}|}}\leq 1$. Suppose also that
there exists constants $c>0$ and $C_{1},C_{2}>0$ such that for any number
field $k$ and any $X\geq 1$, we have for each $i=1,2$ that

$\\#\mathcal{F}_{k}(X;G/N_{i})\leq C_{i}X^{\frac{c}{\sqrt{|G/N_{i}|}}}.$

Then

$\\#\mathcal{F}_{k}(X;G)\leq C_{1}C_{2}X^{\frac{c}{\sqrt{|G|}}}$

for every number field $k$ and every $X\geq 1$.

###### Proof.

By Lemma 2.1 and the assumptions of the lemma, we find

$\\#\mathcal{F}_{k}(X;G)\leq
C_{1}C_{2}X^{\frac{c}{|N_{1}|\sqrt{|G/N_{1}|}}+\frac{c}{|N_{2}|\sqrt{|G/N_{2}|}}}=C_{1}C_{2}X^{\frac{c}{\sqrt{|G|}}\left(\frac{1}{\sqrt{|N_{1}|}}+\frac{1}{\sqrt{|N_{2}|}}\right)}\leq
C_{1}C_{2}X^{\frac{c}{\sqrt{|G|}}},$

as claimed. ∎

We are particularly interested in the case when $N_{1}$ and $N_{2}$ are
distinct minimal normal subgroups, for in this case, the condition that
$N_{1}\cap N_{2}=1$ is automatic. The following lemma records a
characterization of minimal normal subgroups.

######  Lemma 2.3.

Let $G$ be a finite group and $N\trianglelefteq G$ a minimal normal subgroup.
Then either $N\simeq\mathbb{F}_{p}^{r}$ for some prime $p$ and integer $r\geq
1$ or $N\simeq T^{r}$ for some nonabelian simple group $T$ and integer $r\geq
1$.

###### Proof.

This is standard, and follows from the fact that any minimal normal subgroup
must be characteristically simple. See [KS04, 1.7.3], for example. ∎

Notably, if $G$ has two minimal normal subgroups $N_{1}$ and $N_{2}$, at least
one of which is nonabelian, then we must have
$\frac{1}{\sqrt{|N_{1}|}}+\frac{1}{\sqrt{|N_{2}|}}\leq\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{60}}<1$,
since any nonabelian simple group has order at least $60$. From this, we find
the easy but profoundly useful reduction:

######  Lemma 2.4.

Suppose that the conclusions of Theorems 1.3–1.5 hold for every finite group
$G$ such that either:

  * •

$G$ has a unique minimal normal subgroup $N$, and $N$ is nonabelian; or

  * •

all minimal normal subgroups of $G$ are abelian.

Then the conclusions of Theorems 1.3–1.5 hold for all finite groups $G$.

###### Proof.

For Theorems 1.4 and 1.5, this follows directly from Lemma 2.2, Lemma 2.3, and
the above discussion. It also follows for Theorem 1.3 analogously, but with an
additional straightforward computation that we omit. ∎

######  Remark.

If one wishes to show that there is some constant $c>0$ such that
$\\#\mathcal{F}_{k}(X;G)\ll X^{\frac{c}{|G|}}$ for every finite group $G$ (as
would follow from Malle’s conjecture), then these reductions show that it
suffices to prove that such a constant exists for finite groups $G$ with a
unique minimal normal subgroup since we trivially have
$\frac{1}{2}+\frac{1}{2}\leq 1$.

We now split our discussion according to the two cases of Lemma 2.4. We begin
by treating the case that all minimal normal subgroups of $G$ are abelian.

##  3\. Groups with abelian minimal normal subgroups

In this section, we prove Theorems 1.3–1.5 for groups $G$ for which all
minimal normal subgroups are abelian, assuming by way of induction that these
theorems are true for groups of smaller order. To this end, we begin by
recording some necessary bounds on class groups and abelian extensions of
number fields. We then use these results to provide bounds on “central
extensions” of number fields, which will prove to be the key in our treatment
of this case. In their seminal work on nilpotent groups, Klüners and Malle
[KM04] also crucially exploited central extensions, though the mechanics of
their proof are, of necessity, a little different than ours.

###  3.1. Bounds on class groups and the number of abelian extensions of a
number field

We begin by providing a general bound on the class group of a number field. In
situations where the dependence of such a bound on the degree $[k:\mathbb{Q}]$
is irrelevant (as will prove to be the case in the proof of Theorems 1.4 and
1.5), the bound
$|\mathrm{Cl}(k)|\ll_{[k:\mathbb{Q}],\epsilon}|\mathrm{Disc}(k)|^{\frac{1}{2}+\epsilon}$
will be sufficient. However, in situations where the degree dependence is
relevant (e.g., as will be the case in the proof of Theorem 1.3), we will
instead make use of the following fully explicit bound.

######  Lemma 3.1.

Let $k$ be a number field, let $d:=[k:\mathbb{Q}]$, and let $\mathrm{Cl}(k)$
denote the ideal class group of $k$. Then

$|\mathrm{Cl}(k)|\leq 2\pi\cdot|\mathrm{Disc}(k)|^{3/4}.$

###### Proof.

From Louboutin [Lou00, Equation (2)], we find

$\displaystyle|\mathrm{Cl}(k)|$
$\displaystyle\leq\frac{w_{k}}{2R_{k}}\left(\frac{2}{\pi}\right)^{r_{2}}\left(\frac{e\log|\mathrm{Disc}(k)|}{4(n-1)}\right)^{n-1}|\mathrm{Disc}(k)|^{1/2}$
$\displaystyle\leq\frac{w_{k}}{2R_{k}}\left(\frac{2}{\pi}\right)^{r_{2}}|\mathrm{Disc}(k)|^{3/4},$

where $R_{k}$ denotes the regulator of $k$ and $w_{k}$ is the number of roots
of unity in $k$. Appealing to lower bounds on the ratio $\frac{R_{k}}{w_{k}}$
due to Zimmert [Zim81, Satz 3] (taking $\gamma=1.5$), the claim follows unless
$k$ is imaginary quadratic. However, if $k$ is imaginary quadratic, then
$\frac{w_{k}}{2R_{k}}=1$ unless $k=\mathbb{Q}(\zeta_{3})$ or
$k=\mathbb{Q}(i)$, and we observe that the claim follows in all cases. ∎

######  Remark.

The constants $2\pi$ and $3/4$ in the upper bound $|\mathrm{Cl}(k)|\leq
2\pi|\mathrm{Disc}(k)|^{3/4}$ are not optimal in general. For example, as
Louboutin notes, stronger bounds on the class number follow from combining
work of Lenstra [Len92] with that of Zimmert [Zim81, Satz 2]; the constant
$2\pi$ in particular may be replaced by an expression decaying exponentially
with the degree of $k$. For our purposes, however, the exact constants are not
of particular importance, and will have little impact on the final statement
of Theorem 1.3. We have therefore used a relatively simpler statement with the
intent of aiding the clarity of the argument to follow.

We now provide for any abelian group $A$, a general bound on the number of
$A$-extensions of a fixed number field. We begin by bounding the number of
such extensions with a fixed relative discriminant.

######  Lemma 3.2.

Let $A$ be an abelian group, $k$ a number field, and $\mathfrak{D}$ a
squarefree ideal of $k$. Then the number of $A$-extensions of $k$ whose finite
ramified primes are exactly those dividing $\mathfrak{D}$ may be bounded by

$|A|^{3d}\cdot|A[2]|^{r_{1}(k)}\cdot|\mathrm{Hom}(\mathrm{Cl}(k),A)|\cdot\prod_{\mathfrak{p}\mid\mathfrak{D}}(|A|-1),$

where $d=[k:\mathbb{Q}]$, $A[2]$ is the $2$-torsion subgroup of $A$, and
$r_{1}(k)$ is the number of real places of $k$.

Similarly, the number of $A$-extensions of $k$ whose finite ramified primes
divide $\mathfrak{D}$ (but may not include all prime divisors of
$\mathfrak{D}$) may be bounded by

$|A|^{2d}\cdot|A[2]|^{r_{1}(k)}\cdot|\mathrm{Hom}(\mathrm{Cl}(k),A)|\cdot\prod_{\mathfrak{p}\mid\mathfrak{D}}|A|.$

###### Proof.

Let $I_{k}:=\mathbb{A}_{k}/k^{\times}$ be the idèle class group. By class
field theory, $A$-extensions of $k$ correspond to surjective homomorphisms
$I_{k}\to A$. Recall the fundamental exact sequence

$0\to\mathcal{O}_{k}^{\times}\to\prod_{v}\mathcal{O}_{v}^{\times}\to
I_{k}\to\mathrm{Cl}(k)\to 0,$

where the product runs over places $v$ of $k$, and where, for finite $v$,
$\mathcal{O}_{v}^{\times}$ is the unit group of the ring of integers
$\mathcal{O}_{v}$ of the completion $k_{v}$. Since the functor
$\mathrm{Hom}(-,A)$ is left exact, we find the exact sequence

$0\to\mathrm{Hom}(\mathrm{Cl}(k),A)\to\mathrm{Hom}(I_{k},A)\to\prod_{v}\mathrm{Hom}(\mathcal{O}_{v}^{\times},A).$

By continuity, given $\rho\in\mathrm{Hom}(I_{k},A)$, the image of $\rho$ in
$\mathrm{Hom}(\mathcal{O}_{v}^{\times},A)$ can be nontrivial at finitely many
places; in fact, the places for which the map is nontrivial are exactly the
places at which the extension corresponding to $\rho$ is ramified. Hence, the
number of $A$-extensions ramified exactly at finite primes dividing
$\mathfrak{D}$ may be bounded by

$|\mathrm{Hom}(\mathrm{Cl}(k),A)|\cdot\prod_{\mathfrak{p}\mid\mathfrak{D}}\left(|\mathrm{Hom}(\mathcal{O}_{\mathfrak{p}}^{\times},A)|-1\right)\cdot\prod_{v\mid\infty}|\mathrm{Hom}(\mathcal{O}_{v}^{\times},A)|,$

while those that are ramified at most at primes dividing $\mathfrak{D}$ may be
bounded by essentially the same expression, but without the term $-1$ in the
product over finite primes $\mathfrak{p}\mid\mathfrak{D}$. The product over
infinite places $v$ yields the expression $|A[2]|^{r_{1}(k)}$ in the claim.
Thus, it suffices to bound
$|\mathrm{Hom}(\mathcal{O}_{\mathfrak{p}}^{\times},A)|$ for finite primes
$\mathfrak{p}$.

Letting $\pi$ be a uniformizer for $\mathcal{O}_{\mathfrak{p}}$, we have

$0\to
1+\pi\mathcal{O}_{\mathfrak{p}}\to\mathcal{O}_{\mathfrak{p}}^{\times}\to(\mathcal{O}_{\mathfrak{p}}/\mathfrak{p})^{\times}\to
0,$

and so $|\mathrm{Hom}(\mathcal{O}_{\mathfrak{p}}^{\times},A)|$ may be bounded
by the product
$|\mathrm{Hom}(1+\pi\mathcal{O}_{\mathfrak{p}},A)|\cdot|\mathrm{Hom}((\mathcal{O}_{\mathfrak{p}}/\mathfrak{p})^{\times},A)|$.
Since $(\mathcal{O}_{\mathfrak{p}}/\mathfrak{p})^{\times}$ is cyclic, we have
$|\mathrm{Hom}((\mathcal{O}_{\mathfrak{p}}/\mathfrak{p})^{\times},A)|\leq|A|$.
Moreover, since $1+\pi\mathcal{O}_{\mathfrak{p}}$ is a pro-$p$ group (where
$p$ is the rational prime lying below $\mathfrak{p}$),
$\mathrm{Hom}(1+\pi\mathcal{O}_{\mathfrak{p}},A)$ will be trivial unless
$p\mid|A|$.

By means of the standard log map, we also have

$0\to\mu_{p^{\infty}}(k_{\mathfrak{p}})\to
1+\pi\mathcal{O}_{\mathfrak{p}}\to\mathcal{O}_{\mathfrak{p}}\to 0,$

where $\mu_{p^{\infty}}$ is the set of $p$-power roots of unity, so that
$|\mathrm{Hom}(1+\pi\mathcal{O}_{\mathfrak{p}},A)|\leq|A_{p}|\cdot|\mathrm{Hom}(\mathcal{O}_{\mathfrak{p}},A_{p})$,
where $A_{p}$ is the Sylow $p$-subgroup of $A$. Finally, since
$\mathcal{O}_{\mathfrak{p}}\simeq\mathbb{Z}_{p}^{f_{\mathfrak{p}}}$ as an
additive group (where
$f_{\mathfrak{p}}:=[\mathcal{O}_{\mathfrak{p}}/\mathfrak{p}:\mathbb{F}_{p}]$),
we find
$|\mathrm{Hom}(\mathcal{O}_{\mathfrak{p}},A_{p})|\leq|A_{p}|^{f_{\mathfrak{p}}}$.
Hence, for primes $\mathfrak{p}$ dividing $|A|$, we obtain

$|\mathrm{Hom}(\mathcal{O}_{\mathfrak{p}}^{\times},A)|\leq|A|\cdot|A_{p}|^{1+f_{\mathfrak{p}}}\leq(|A|-1)\cdot|A_{p}|^{3f_{\mathfrak{p}}}.$

Thus, all told, we see that the number of $A$-extensions whose finite ramified
primes are exactly those dividing $\mathfrak{D}$ is bounded by

$|A[2]|^{r_{1}(k)}\cdot|\mathrm{Hom}(\mathrm{Cl}(k),A)|\cdot\prod_{\mathfrak{p}\mid\mathfrak{D}}(|A|-1)\cdot\prod_{\mathfrak{p}\mid|A|}|A_{p}|^{3f_{\mathfrak{p}}}.$

Since for any rational prime $p$, we have $\sum_{\mathfrak{p}\mid
p}f_{\mathfrak{p}}=d$, the first result follows. The second follows
analogously. ∎

######  Remark.

Note that, particularly in the second case of Lemma 3.2, we have not enforced
the condition that the element $\rho\in\mathrm{Hom}(I_{k},A)$ is surjective.
As a result, Lemma 3.2 in fact furnishes us with a bound on the number of
$A_{0}$-extensions of $k$ whose finite ramified primes divide $\mathfrak{D}$
across all subgroups $A_{0}\leq A$ simultaneously. We will exploit this fact
later.

Using this, we now obtain a bound on the set $\\#\mathcal{F}_{k}(X;A)$ for any
abelian group $A$ and any number field $k$.

######  Lemma 3.3.

Let $k$ be a number field, and let $A$ be an abelian group of rank $r\geq 1$.
Let $p$ be the smallest prime divisor of $|A|$, and set $a=\frac{p-1}{p}|A|$
and $m=d(|A|-1)$. Then for any $X\geq 1$, we have

$\\#\mathcal{F}_{k}(X;A)\leq\frac{|A|^{3d}\cdot|A[2]|^{r_{1}(k)}}{(m-1)!}|\mathrm{Cl}(k)|^{r}X^{\frac{1}{a}}(\log
X^{\frac{1}{a}}+m-1)^{m-1}|\mathrm{Disc}(k)|^{-\frac{p}{p-1}},$

and in particular also

$\\#\mathcal{F}_{k}(X;A)\leq|A|^{3d}\cdot|A[2]|^{r_{1}(k)}\cdot
e^{d(|A|-1)-1}\cdot|\mathrm{Cl}(k)|^{r}\cdot
X^{\frac{2}{a}}\cdot|\mathrm{Disc}(k)|^{-\frac{p}{p-1}}.$

###### Proof.

We first observe that for any $K\in\mathcal{F}_{k}(X;A)$, the relative
discriminant ideal $\mathfrak{D}_{K/k}$ must be $a$-powerful (i.e., the
valuation $v_{\mathfrak{p}}(\mathfrak{D}_{K/k})$ must be at least $a$ for any
prime $\mathfrak{p}$ dividing $\mathfrak{D}_{K/k}$). As a result, if we let
$\mathfrak{D}$ be the product of the (finite) primes ramified in $K$, then we
must have
$|\mathfrak{D}|\leq(X/|\mathrm{Disc}(k)|^{|A|})^{\frac{1}{a}}=X^{\frac{1}{a}}|\mathrm{Disc}(k)|^{-\frac{p}{p-1}}$,
where $|\mathfrak{D}|$ denotes the ideal norm of $\mathfrak{D}$. Hence, by
Lemma 3.2, we find

$\displaystyle\\#\mathcal{F}_{k}(X;A)$
$\displaystyle\leq|A|^{3d}|A[2]|^{r_{1}(k)}|\mathrm{Hom}(\mathrm{Cl}(k),A)|\sum_{|\mathfrak{D}|\leq
X^{\frac{1}{a}}|\mathrm{Disc}(k)|^{-\frac{p}{p-1}}}\tau_{|A|-1}(\mathfrak{D})$
$\displaystyle\leq|A|^{3d}|A[2]|^{r_{1}(k)}|\mathrm{Hom}(\mathrm{Cl}(k),A)|\sum_{D\leq
X^{\frac{1}{a}}|\mathrm{Disc}(k)|^{-\frac{p}{p-1}}}\tau_{d(|A|-1)}(D),$

where the final summation runs over positive integers $D$, and where, for any
$m\geq 1$, $\tau_{m}$ denotes the $m$-fold divisor function. We note that
since $\mathrm{Cl}(k)$ is abelian and $A$ is assumed to have rank $r$, we have
$|\mathrm{Hom}(\mathrm{Cl}(k),A)|\leq|\mathrm{Cl}(k)|^{r}$. Meanwhile, by
[Bor02], for any integer $m\geq 1$ and any $Q\geq 1$, we have

$\sum_{D\leq Q}\tau_{m}(D)\leq\frac{Q}{(m-1)!}(\log Q+m-1)^{m-1},$

and this yields the first claim.

For the second, we use the standard inequalities
$(m-1)!\geq\left(\frac{m-1}{e}\right)^{m-1}$ and $\log(1+x)\leq x$ to find
that

$\frac{1}{(m-1)!}\left(\log X^{\frac{1}{a}}+m-1\right)^{m-1}\leq
e^{m-1}\left(\frac{\log X}{a(m-1)}+1\right)^{m-1}\leq e^{m-1}X^{\frac{1}{a}}.$

This completes the proof. ∎

Using the first case of Lemma 3.3, we also obtain the following inexplicit
bound.

######  Corollary 3.4.

Let $k$ be a number field and let $A$ be an abelian group of rank $r\geq 1$.
Let $p$ be the smallest prime divisor of $|A|$ and set $a:=\frac{p-1}{p}|A|$.
Then for any $X\geq 1$ and any $\epsilon>0$, we have

$\\#\mathcal{F}_{k}(X;A)\ll_{[k:\mathbb{Q}],|A|,\epsilon}|\mathrm{Cl}(k)|^{r}X^{\frac{1}{a}+\epsilon}|\mathrm{Disc}(k)|^{-\frac{p}{p-1}}.$

###### Proof.

This follows from the first inequality in Lemma 3.3. ∎

###  3.2. Bounds on central extensions

Recall that a subgroup $A$ of a (finite) group $G$ is called _central_ if it
is contained in the center $Z(G)$. Such a subgroup $A$ is necessarily a normal
abelian subgroup of $G$, and the aim of this section is to provide useful
bounds on $\\#\mathcal{F}_{k}(X;G)$ in terms of $\\#\mathcal{F}_{k}(X;G/A)$ in
the case that $G$ has a nontrivial central subgroup $A$. The key fact we use
in this case is that $G$-extensions of a number field $k$ with the same
$G/A$-subextension “differ” by $A$-extensions of the base field $k$. This is
made precise by the following lemma.

######  Lemma 3.5.

Let $G$ be a finite group and let $A\leq Z(G)$ (which implies that $A$ is
abelian and normal in $G$). Let $k$ be a number field and
$G_{k}:=\mathrm{Gal}(\overline{k}/k)$ the absolute Galois group of $k$. Let
$\rho\colon G_{k}\rightarrow G$ be a surjective homomorphism, and let
$\bar{\rho}\colon G_{k}\to G/A$ be the projection of $\rho$.

Then the set of homomorphisms $\rho^{\prime}\colon G_{k}\to G$ whose
projection $\bar{\rho^{\prime}}$ equals $\bar{\rho}$ is in one-to-one
correspondence with the set of homomorphisms $\psi\colon G_{k}\to A$.

###### Proof.

Let $\rho$ and $\rho^{\prime}$ be as in the statement of the lemma. We
initially define a map $\psi\colon G_{k}\to A$ by
$\psi(\sigma)=\rho^{\prime}(\sigma)\rho(\sigma)^{-1}$ whose image lands in $A$
by the assumption that $\bar{\rho^{\prime}}=\bar{\rho}$. Since $A$ is central,
we then find
$\psi(\sigma_{1}\sigma_{2})=\rho^{\prime}(\sigma_{1})\rho^{\prime}(\sigma_{2})\rho(\sigma_{2})^{-1}\rho(\sigma_{1})^{-1}=\rho^{\prime}(\sigma_{1})\rho(\sigma_{1})^{-1}\rho^{\prime}(\sigma_{2})\rho(\sigma_{2})^{-1}=\psi(\sigma_{1})\psi(\sigma_{2})$,
so that $\psi$ is in fact a homomorphism. This defines the correspondence in
one direction. For the other, we simply take $\rho^{\prime}$ to be the product
$\rho\psi$. ∎

Using this, we obtain the following pair of lemmas.

######  Lemma 3.6.

Let $G$ be a finite group and let $A\leq Z(G)$. Let $k$ be a number field, let
$F/k$ be a $G/A$-extension, and let $\mathfrak{D}$ be a squarefree ideal of
$k$ divisible by every finite prime ramified in $F$. Then the number of
$G$-extensions $K/k$ for which $K^{A}=F$ and whose finite ramified primes are
exactly those dividing $\mathfrak{D}$ may be bounded by

$e^{\frac{(\log|G|)^{2}}{\log
2}}\cdot|A|^{2d}\cdot|A[2]|^{r_{1}(k)}\cdot|\mathrm{Hom}(\mathrm{Cl}(k),A)|\cdot\prod_{\mathfrak{p}\mid\mathfrak{D}}|A|,$

where $d=[k:\mathbb{Q}]$.

###### Proof.

We first observe that any two surjective homomorphisms
$\bar{\rho}_{0},\bar{\rho}_{1}\colon G_{k}\to G/A$ for which
$\overline{k}^{\ker\bar{\rho}_{0}}=\overline{k}^{\ker\bar{\rho}_{1}}=F$ must
differ by post-composition of an automorphism of $G/A$. In particular, there
are at most $|\mathrm{Aut}(G/A)|$ choices for the projection $\bar{\rho}\colon
G_{k}\to G/A$ of a surjective homomorphism $\rho\colon G_{k}\to G$ for which
$K=\overline{k}^{\ker\rho}$ satisfies $K^{A}=F$. We fix a choice of projection
$\bar{\rho}$, noting that since $G/A$ is generated by at most $\Omega(|G/A|)$
elements, $|\mathrm{Aut}(G/A)|\leq|G|^{\Omega(|G/A|)}$, so there are at most
this many choices of the projection $\bar{\rho}$.

We now bound the number of surjective homomorphisms $\rho^{\prime}\colon
G_{k}\to G$ such that $\bar{\rho^{\prime}}=\bar{\rho}$ and
$\overline{k}^{\ker\rho^{\prime}}$ is ramified exactly at $\mathfrak{D}$. We
may suppose there is at least one such $\rho^{\prime}$ (which by an abuse of
notation, we denote as $\rho$) such that $\overline{k}^{\ker\rho}$ is ramified
exactly at $\mathfrak{D}$. (If not, the class $\bar{\rho}$ will contribute
nothing to the count.) By Lemma 3.5, we must have $\rho^{\prime}=\rho\psi$ for
some $\psi\colon G_{k}\to A$. Let $A_{0}\leq A$ denote the image of $\psi$. In
order for $\overline{k}^{\ker\rho^{\prime}}$ to be ramified only at
$\mathfrak{D}$, it follows that the $A_{0}$-extension of $k$ corresponding to
$\psi$ may be ramified only at primes dividing $\mathfrak{D}$. The result then
follows from Lemma 3.2 and the remark following its proof, together with the
fact that there are at most $|A_{0}|^{\Omega(|A_{0}|)}\leq|G|^{\Omega(|A|)}$
choices for $\psi$ corresponding to a given extension
$\overline{k}^{\ker\psi}$. Noting that
$|G|^{\Omega(|G/A|)+\Omega(|A|)}=|G|^{\Omega(|G|)}\leq
e^{\frac{(\log|G|)^{2}}{\log 2}}$, the result follows. ∎

######  Lemma 3.7.

Let $G$ be a finite group and let $A\leq Z(G)$ have rank $r$. Let $k$ be a
number field, and suppose that $F/k$ is a $G/A$-extension. Define
$a=\frac{p-1}{p}|G|$, where $p$ is the least prime dividing $|G|$. Let
$m=d|A|$. Then for any $X\geq 1$, we have

$\displaystyle\\#\\{K\in\mathcal{F}_{k}(X;G):K^{A}=F\\}$
$\displaystyle\quad\quad\quad\leq\frac{e^{\frac{(\log|G|)^{2}}{\log
2}}\cdot|A|^{2d}\cdot|A[2]|^{r_{1}(k)}}{(m-1)!}\cdot\frac{|\mathrm{Cl}(k)|^{r}}{|\mathrm{Disc}(k)|^{-\frac{p}{p-1}}}\cdot
X^{\frac{1}{a}}\cdot(\log X^{\frac{1}{a}}+m-1)^{m-1},$

and hence also

$\\#\\{K\in\mathcal{F}_{k}(X;G):K^{A}=F\\}\leq e^{d|A|-1}\cdot
e^{\frac{(\log|G|)^{2}}{\log
2}}\cdot|A|^{2d}\cdot|A[2]|^{r_{1}(k)}\cdot|\mathrm{Cl}(k)|^{r}\cdot
X^{\frac{2}{a}}|\mathrm{Disc}(k)|^{-\frac{p}{p-1}}$

and

$\\#\\{K\in\mathcal{F}_{k}(X;G):K^{A}=F\\}\ll_{d,G,\epsilon}|\mathrm{Cl}(k)|^{r}X^{\frac{1}{a}+\epsilon}|\mathrm{Disc}(k)|^{-\frac{p}{p-1}}.$

###### Proof.

Since the relative discriminant ideal $\mathfrak{D}_{K/k}$ of any extension
$K\in\mathcal{F}_{k}(X;G)$ must be $a$-powerful, by Lemma 3.6, we find

$\displaystyle\\#\\{K\in\mathcal{F}_{k}(X;G):K^{A}=F\\}$
$\displaystyle\quad\quad\leq e^{\frac{(\log|G|)^{2}}{\log
2}}\cdot|A|^{2d}\cdot|A[2]|^{r_{1}(k)}\cdot|\mathrm{Hom}(\mathrm{Cl}(k),A)|\cdot\sum_{\begin{subarray}{c}\mathfrak{D}:\\\
|\mathfrak{D}|\leq(X/\mathrm{Disc}(k)^{|G|})^{\frac{1}{a}}\end{subarray}}\tau_{|A|}(\mathfrak{D}).$

Proceeding as in the proof of Lemma 3.3, the result follows. ∎

######  Remark.

While we find Lemma 3.7 convenient for our purposes, we note that there should
be versions of Lemma 3.7 that obtain a savings in terms of the discriminant of
the field $F=K^{A}$. However, such arguments must also account for the wild
parts of the discriminants of $K$ and $F$, which potentially have a rather
larger dependence on the parameters $d$ and $G$ than we care to allow here.
For example, see [Lem23, Lemma 3.10] for bounds on the wild part of the
discriminant.

###  3.3. Proof of Theorems 1.3–1.5 when $G$ has an abelian minimal normal
subgroup

We are now ready to prove Theorems 1.3–1.5 in the case that $G$ has an abelian
minimal normal subgroup $N$. We may assume by way of induction that the
results are known for all groups of order smaller than $G$, in particular any
quotient or subgroup of $G$, and over any number field.

We begin with the proof of Theorem 1.4, as we believe it most cleanly conveys
the key ideas. This proof will also essentially at the same time provide a
proof of Theorem 1.5 for reasons to be explained, and will provide a template
on which the proof of Theorem 1.3 will be based.

###### Proof of Theorem 1.4 when $G$ has an abelian minimal normal subgroup
$N$.

Let $G$ be a finite group, all of whose minimal normal subgroups are abelian.
We suppose by way of induction that there is some constant $c_{0}>0$ such that
for every group $G_{0}$ of order less than $|G|$, every number field $k$, and
every $X\geq 1$, there holds

$\\#\mathcal{F}_{k}(X;G_{0})\ll_{[k:\mathbb{Q}],G_{0}}X^{\frac{c_{0}}{\sqrt{|G_{0}|}}}.$

We consider the constant $c_{0}$ in some sense as a parameter, with the goal
of observing the conditions on $c_{0}$ that must hold. Theorem 1.4 will follow
if we show that $c_{0}=4$ is admissible.

Note that if $G$ is abelian of rank at most $2$, then Corollary 3.4 shows that

(3.1)
$\\#\mathcal{F}_{k}(X;G)\ll_{[k:\mathbb{Q}],G,\epsilon}X^{\frac{2}{|G|}+\epsilon}\ll_{[k:\mathbb{Q}],G,\epsilon}X^{\frac{\sqrt{2}}{\sqrt{|G|}}+\epsilon},$

which in particular is consistent with any $c_{0}>\sqrt{2}$, and may be
regarded as a base case for this part of the induction.

By Lemma 2.3, there is some prime $p$ and some integer $r\geq 1$ so that
$N\simeq\mathbb{F}_{p}^{r}$. Let $C=C_{G}(N)$ be the centralizer in $G$ of
$N$, and note that $G/C$ is naturally a subgroup of $\mathrm{Aut}(N)$. We
consider three possibilities in turn:

  1. Case 1)

$r=1$, so that $N\simeq\mathbb{F}_{p}$;

  2. Case 2)

$r\geq 2$ and $|N|\geq\sqrt{|G|}$; and

  3. Case 3)

$r\geq 2$ and $|N|\leq\sqrt{|G|}$.

It evidently suffices to prove the theorem in these three cases.

_Case 1)_ Assume that $N\simeq\mathbb{F}_{p}$. We first consider the
possibility that $C=G$, i.e. that $N$ is central in $G$. Note that this must
be the case if $p=2$. By Lemma 3.7, we find

$\\#\mathcal{F}_{k}(X;G)\ll_{[k:\mathbb{Q}],G,\epsilon}X^{\frac{p_{0}}{(p_{0}-1)|G|}}\\#\mathcal{F}_{k}(X^{1/p};G/N)\ll_{[k:\mathbb{Q}],G,\epsilon}X^{\frac{p_{0}}{(p_{0}-1)|G|}+\frac{c_{0}}{\sqrt{p|G|}}+\epsilon}$

where $p_{0}$ is the least prime dividing $|G|$. We have
$\frac{p_{0}}{p_{0}-1}\leq 2$ and $p\geq 2$, so it suffices to restrict to
$c_{0}$ satisfying

$\frac{2}{\sqrt{|G|}}+\frac{c_{0}}{\sqrt{2}}<c_{0},\quad\text{i.e.}\quad
c_{0}<\frac{4+2\sqrt{2}}{\sqrt{|G|}}.$

This is compatible with any $c_{0}>3$ once $|G|\geq 6$. However, if $|G|\leq
5$, then $G$ must be abelian, and we may appeal to (3.1) instead.

We next consider the possibility that $C=N$ and that
$G/C\simeq\mathrm{Aut}(N)\simeq\mathbb{F}_{p}^{\times}$, i.e. that
$G\simeq\mathbb{F}_{p}\rtimes\mathbb{F}_{p}^{\times}$. In this case, we have

$\displaystyle\\#\mathcal{F}_{k}(X;G)$
$\displaystyle\leq\sum_{F\in\mathcal{F}_{k}(X^{1/p};C_{p-1})}\\#\mathcal{F}_{F}(X;C_{p})$
$\displaystyle\ll_{[k:\mathbb{Q}],p,\epsilon}X^{\frac{1}{p-1}+\epsilon}\sum_{F\in\mathcal{F}_{k}(X^{1/p};C_{p-1})}|\mathrm{Disc}(F)|^{\frac{1}{2}-\frac{p}{p-1}}$
$\displaystyle\ll_{[k:\mathbb{Q}],p,\epsilon}X^{\frac{1}{p-1}+\epsilon}.$

Since $|G|=p(p-1)$ in this case, this is consistent with any
$c_{0}>\sqrt{\frac{3}{2}}\approx 1.224$, which is sufficient.

We may therefore assume that $p\geq 3$ and that either $|C|\geq 2|N|=2p$ or
$|G/C|\leq\frac{p-1}{2}$. These latter two conditions imply that we must have
$|G|\leq\frac{1}{2}|C|^{2}$, and hence that $|C|\geq\sqrt{2|G|}$. We also have
the bound $|C|\geq\frac{1}{p-1}|G|$ since $[G:C]\leq p-1$. Appealing to Lemma
3.7 and the induction hypothesis, we find

$\displaystyle\\#\mathcal{F}_{k}(X;G)$
$\displaystyle\leq\sum_{F\in\mathcal{F}_{k}(X^{\frac{1}{|N|}};G/N)}\\#\\{K\in\mathcal{F}_{F^{C/N}}(X;C):K^{N}=F\\}$
$\displaystyle\ll_{[k:\mathbb{Q}],G,\epsilon}X^{\frac{p_{0}}{(p_{0}-1)|C|}+\epsilon}\cdot\\#\mathcal{F}_{k}(X^{\frac{1}{|N|}};G/N)$
$\displaystyle\ll_{[k:\mathbb{Q}],G,\epsilon}X^{\frac{2}{|C|}+\frac{c_{0}}{\sqrt{|G||N|}}+\epsilon},$

where $p_{0}$ is the least prime dividing $|C|$. When $p\geq 5$, we use the
bounds $|C|\geq\sqrt{2|G|}$ and $|N|\geq 5$ to conclude this is consistent
with the desired bound
$O_{[k:\mathbb{Q}],G,\epsilon}(X^{\frac{c_{0}}{\sqrt{|G|}}+\epsilon})$ for any
$c_{0}>0$ satisfying

$\sqrt{2}+\frac{c_{0}}{\sqrt{5}}<c_{0},\quad\text{i.e.}\quad
c_{0}>\frac{5\sqrt{2}+\sqrt{10}}{4}\approx 2.558.$

When $p=3$, we use the bound $|C|\geq|G|/2$, and see that this is compatible
with the desired bound when

$\frac{4}{\sqrt{|G|}}+\frac{c_{0}}{\sqrt{3}}<c_{0},\quad\text{i.e.}\quad
c_{0}>\frac{2(3+\sqrt{3})}{\sqrt{|G|}}.$

Thus, any $c_{0}>3$ is admissible when $|G|\geq 12$. When $|G|\leq 9$ and $G$
is abelian, we may use (3.1). Finally, the nonabelian group of order $6$ is
isomorphic to $\mathbb{F}_{3}\rtimes\mathbb{F}_{3}^{\times}$, a case that was
treated earlier. This completes the proof of Case 1, the upshot of which is
that this case is consistent with any constant $c_{0}>3$ arising from the
induction hypothesis.

_Case 2)_ Here, we are supposing that $N\simeq\mathbb{F}_{p}^{r}$ for some
$r\geq 2$ and that $|N|\geq\sqrt{|G|}$. Let $W\leq N$ be a subspace of
codimension $1$, so that $W\simeq\mathbb{F}_{p}^{r-1}$. As $N$ is abelian, $W$
is normal in $N$, but since we have assumed that $N$ is minimal, it follows
that $W$ is corefree in $G$. (I.e., that $\cap_{g\in G}W^{g}=1$.) In
particular, any $K\in\mathcal{F}_{k}(X;G)$ is determined by its subfield
$K^{W}$, which is a cyclic extension of $K^{N}$ with degree $p$. Moreover,
$|\mathrm{Disc}(K^{W})|\leq|\mathrm{Disc}(K)|^{\frac{1}{|W|}}\leq
X^{\frac{p}{|N|}}$. Appealing to Corollary 3.4 to bound the number of
possibilities for the extension $K^{W}/K^{N}$, we therefore find by our
induction hypothesis that

$\displaystyle\\#\mathcal{F}_{k}(X;G)$
$\displaystyle\leq\sum_{F\in\mathcal{F}_{k}(X^{\frac{1}{|N|}};G/N)}\\#\mathcal{F}_{F}(X^{\frac{p}{|N|}};C_{p})$
$\displaystyle\ll_{[k:\mathbb{Q}],G,\epsilon}X^{\frac{p}{(p-1)|N|}+\epsilon}\sum_{F\in\mathcal{F}_{k}(X^{\frac{1}{|N|}};G/N)}|\mathrm{Disc}(F)|^{\frac{1}{2}-\frac{p}{p-1}}$
$\displaystyle\ll_{[k:\mathbb{Q}],G,\epsilon}X^{\frac{p}{(p-1)|N|}+\max\left\\{0,\frac{c_{0}}{\sqrt{|G||N|}}+\frac{1}{2|N|}-\frac{p}{(p-1)|N|}\right\\}+\epsilon}.$

If the maximum in the exponent above is $0$, then we find using our assumption
that $|N|\geq\sqrt{|G|}$ that

$\frac{p}{(p-1)|N|}\leq\frac{2}{\sqrt{|G|}},$

which is consistent with the bound
$O_{[k:\mathbb{Q}],G,\epsilon}(X^{\frac{c_{0}}{\sqrt{|G|}}})$ for any
$c_{0}>2$. If the maximum is not $0$, then the bound above becomes
$O_{[k:\mathbb{Q}],G,\epsilon}(X^{\frac{c_{0}}{\sqrt{|G||N|}}+\frac{1}{2|N|}+\epsilon})$.
Using our assumptions in this case, together with the lower bound $|N|\geq 4$
that must hold if $r\geq 2$, we find that this is consistent with our goal
provided that

$\frac{c_{0}}{2}+\frac{1}{2}<c_{0},\quad\text{i.e.}\quad c_{0}>1.$

This completes the proof of the second case, with the upshot being that the
induction argument follows provided we assume only that $c_{0}>2$.

_Case 3)_ As in Case 2), let $W\leq N$ again be a subspace of codimension $1$.
Let $N_{G}(W)$ denote its normalizer in $G$, which satisfies
$[G:N_{G}(W)]\leq\frac{p^{r}-1}{p-1}<\frac{|N|}{p-1}$. Observe that $N/W$ is a
normal subgroup of $N_{G}(W)/W$ isomorphic to $\mathbb{F}_{p}$. Let $H\leq
N_{G}(W)$ be the subgroup mapping to its centralizer in the quotient, that is,
$H:=W.C_{N_{G}(W)/W}(N/W)$. We have $[N_{G}(W):H]\leq p-1$, and hence
$[G:H]<|N|$. By construction, $H$ contains $N$, so for any
$K\in\mathcal{F}_{k}(X;G)$, the subfield $K^{H}$ is contained in, and
determined by, $K^{N}$. Additionally, as in Case 2), the extension $K$ will be
determined by its subextension $K^{W}$, which is a cyclic degree $p$ extension
of $K^{N}$. But unlike Case 2), we now exploit the fact that $K^{W}$ is also a
Galois extension of $K^{H}$, with $\mathrm{Gal}(K^{W}/K^{N})\simeq N/W$ a
central subgroup. Note that $|\mathrm{Disc}(K^{W})|\leq X^{\frac{p}{|N|}}$.

In particular, appealing to Lemma 3.7 and letting $p_{0}$ be the least prime
dividing the order of $H/W$, we find

$\displaystyle\\#\mathcal{F}_{k}(X;G)$
$\displaystyle\leq\sum_{F\in\mathcal{F}_{k}(X^{\frac{1}{|N|}};G/N)}\\#\\{K\in\mathcal{F}_{F^{H/N}}(X^{\frac{p}{|N|}};H/W):K^{N/W}=F\\}$
$\displaystyle\ll_{[k:\mathbb{Q}],G,\epsilon}X^{\frac{c_{0}}{\sqrt{|G|}\sqrt{|N|}}+\frac{pp_{0}}{(p_{0}-1)|N||H/W|}+\epsilon}$
$\displaystyle\ll_{[k:\mathbb{Q}],G,\epsilon}X^{\frac{c_{0}}{\sqrt{|G|}\sqrt{|N|}}+\frac{2}{|H|}+\epsilon}.$

Since we have $|H|\geq\frac{|G|}{|N|}\geq\sqrt{|G|}$, a computation as above
reveals this is sufficient to obtain any $c_{0}>3$ if $|N|\geq 9$. If $|N|\leq
8$, we instead directly use the bound $|H|\geq\frac{|G|}{|N|}$. This will be
sufficient for the induction argument provided that

$\frac{c_{0}}{\sqrt{|N|}}+\frac{2|N|}{\sqrt{|G|}}<c_{0},\quad\text{i.e.}\quad
c_{0}>\frac{2|N|}{\sqrt{|G|}}\left(1-\frac{1}{\sqrt{|N|}}\right)^{-1}.$

We then observe that if $|N|=4$, this is sufficient to obtain any $c_{0}>3$
for $|G|\geq 32$, and if $|N|=8$, this is sufficient for $|G|\geq 72$. In
fact, if $|N|=8$, then the condition that $|G|\geq 72$ must be satisfied,
since we have assumed that $|N|\leq\sqrt{|G|}$, which implies that $|G|\geq
64$. However, each group of order $64$ is a $2$-group, hence each of its
minimal normal subgroups will be of order $2$, not $8$. Thus, it remains to
consider the case that $|N|=4$. Here, we must consider groups of order $20$,
$24$, and $28$. Note that we may assume that $N$ is the unique minimal normal
subgroup, since any other minimal normal subgroup will either have order at
most $3$ (hence we may treat $G$ instead by Case 1), with $r=1$) or will have
order $\geq 4$ (in which case Lemma 2.2 implies the desired result). It thus
follows that the only group to consider is the symmetric group $S_{4}$. Here,
we observe that

$\displaystyle\\#\mathcal{F}_{k}(X;S_{4})$
$\displaystyle\leq\sum_{F\in\mathcal{F}_{k}(X^{1/4};S_{3})}\\#\mathcal{F}_{F}(X^{1/2};C_{2})$
$\displaystyle\ll_{[k:\mathbb{Q}],\epsilon}X^{\frac{1}{2}+\epsilon}\sum_{F\in\mathcal{F}_{k}(X^{1/4};S_{3})}|\mathrm{Disc}(F)|^{-\frac{3}{2}}$
$\displaystyle\ll_{[k:\mathbb{Q}],\epsilon}X^{\frac{1}{2}+\epsilon},$

where the last inequality follows, for example, on using with $p=3$ the bound
$\\#\mathcal{F}_{k}(X;\mathbb{F}_{p}\rtimes\mathbb{F}_{p}^{\times})\ll_{[k:\mathbb{Q}],p,\epsilon}X^{\frac{1}{p-1}+\epsilon}$
proved earlier. This is sufficient for any $c_{0}>\sqrt{6}\approx 2.449$, and
concludes the proof of the theorem. ∎

###### Proof of Theorem 1.5 in the case that $G$ has abelian minimal normal
subgroups.

The proof
above shows that any exponent $c_{0}>3$ may be obtained in this step of the
induction argument, so this follows mutatis mutandis from the above proof. ∎

###### Proof of Theorem 1.3 in the case that $G$ has abelian minimal normal
subgroups.

Let $G$ be a finite group, and let $N\simeq\mathbb{F}_{p}^{r}$ be an abelian
minimal normal subgroup of $G$. We follow the same strategy, and use the same
notation, as the proof above, but with the modified induction hypothesis that
there are constants $c_{0},c_{1}$ such that for every group $G_{0}$ of order
less than $|G|$, every number field $k$, and every $X\geq 1$,

$\\#\mathcal{F}_{k}(X;G_{0})\leq
e^{d|G_{0}|}(2d|G_{0}|^{2})^{c_{1}d|G_{0}|^{1/2}}\cdot
X^{\frac{c_{0}}{\sqrt{|G_{0}|}}},$

with the aim of showing that the same bound holds for $G$. As above, we will
track the inequalities that must be satisfied by the parameters $c_{1}$ and
$c_{0}$, with the goal of showing that the claimed values suffice. We first
note that if $G=N=C_{p}$, then Lemma 3.3 and Lemma 3.1 yield

$\\#\mathcal{F}_{k}(X;C_{p})\leq 2\pi\cdot
e^{dp-d-1}2^{d}p^{3d}X^{\frac{2}{p-1}}\leq
e^{dp}(2p^{2})^{3d/2}X^{\frac{2}{p-1}}.$

This is consistent with the induction hypothesis for any $c_{0}\geq
2\sqrt{2}\approx 2.828$ and any $c_{1}\geq\frac{3}{2\sqrt{2}}\approx 1.06$. In
particular, we may assume that $G\neq C_{p}$ below.

_Case 1)_ We begin by assuming that $r=1$, so that $N=\mathbb{F}_{p}$. If $N$
is central, then we may appeal directly to Lemma 3.7 and Lemma 3.1 with our
induction hypothesis to see that

$\displaystyle\\#\mathcal{F}_{k}(X;G)$ $\displaystyle\leq
e^{dp-1}e^{\frac{(\log|G|)^{2}}{\log 2}}2^{d}p^{2d}\cdot 2\pi\cdot
X^{\frac{2p_{0}}{p_{0}|G|}}\cdot\\#\mathcal{F}_{k}(X^{1/p};G/N)$
$\displaystyle\leq e^{dp+d|G|/p}\cdot\frac{2\pi}{e}\cdot
2^{d+c_{1}d|G|^{1/2}p^{-1/2}}\cdot d^{c_{1}d|G|^{1/2}p^{-1/2}}\cdot$
$\displaystyle\quad\quad\cdot(|G|^{2})^{\frac{\log|G|}{2\log
2}+c_{1}d|G|^{1/2}p^{-1/2}}\cdot
X^{\frac{2p_{0}}{(p_{0}-1)|G|}+\frac{c_{0}}{\sqrt{p|G|}}}$ $\displaystyle\leq
e^{d|G|}\cdot(2d|G|^{2})^{c_{1}d|G|^{1/2}}\cdot X^{\frac{c_{0}}{\sqrt{|G|}}}$

with

$c_{1}\geq\left(\frac{1}{2}+\frac{\log(2\pi/e)}{2\log
2}\right)\left(1-\frac{1}{\sqrt{2}}\right)^{-1}\approx 3.770,$

and with any $c_{0}\geq\frac{4}{3}(\sqrt{6}+\sqrt{3})\approx 5.575$ provided
that $|G|\geq 6$. However, if $|G|=4$, then Lemma 3.3 yields a stronger bound
than these $c_{1}$ and $c_{0}$ provide.

We now consider $G\simeq\mathbb{F}_{p}\rtimes\mathbb{F}_{p}^{\times}$ for $p$
odd. Using Lemma 3.3, we find

$\\#\mathcal{F}_{k}(X;\mathbb{F}_{p}\rtimes\mathbb{F}_{p}^{\times})\leq
e^{dp(p-1)-1}p^{3d(p-1)}\cdot 2\pi\cdot
X^{\frac{2}{p-1}}\sum_{F\in\mathcal{F}_{k}(X^{1/p};C_{p-1})}|\mathrm{Disc}(F)|^{\frac{3}{4}-\frac{p}{p-1}}.$

We treat the sum over $F$ via partial summation, splitting into three cases
according to whether $p<13$, $p>13$, or $p=13$. When $p<13$, using partial
summation and Lemma 3.3, we find

$\displaystyle\\#\mathcal{F}_{k}(X;\mathbb{F}_{p}\rtimes\mathbb{F}_{p}^{\times})$
$\displaystyle\leq
e^{dp(p-1)-1}p^{3d(p-1)}\cdot(2\pi)^{2}e^{d(p-1)-1}2^{d}(p-1)^{3d}\cdot\frac{16}{13-p}\cdot
X^{\frac{2}{p-1}+\frac{13-p}{4p(p-1)}}$ $\displaystyle\leq
e^{dp(p-1)}\left(p(p-1)\right)^{3d(p-1)}(p-1)^{-3d(p-2)}\cdot\frac{(2\pi)^{2}}{e^{2d+2}}\cdot
2^{d}e^{dp}\cdot\frac{16}{13-p}\cdot X^{\frac{13+7p}{4p(p-1)}}$
$\displaystyle\leq
e^{d|G|}(2d|G|^{2})^{c_{1}d|G|^{1/2}}X^{\frac{c_{0}}{\sqrt{|G|}}}$

for any $c_{1}\geq\frac{1}{\sqrt{6}}+\frac{3}{\sqrt{6}\cdot\log 2}\approx
2.175$ and any $c_{0}\geq\frac{17}{2\sqrt{6}}\approx 3.470$. Similarly, if
$p>13$, we find

$\displaystyle\\#\mathcal{F}_{k}(X;\mathbb{F}_{p}\rtimes\mathbb{F}_{p}^{\times})$
$\displaystyle\leq
e^{dp(p-1)-1}p^{3d(p-1)}\cdot(2\pi)^{2}e^{d(p-1)-1}2^{d}(p-1)^{3d}\cdot\frac{p+3}{p-13}\cdot
X^{\frac{2}{p-1}}$ $\displaystyle\leq
e^{d|G|}(2d|G|^{2})^{c_{1}d|G|^{1/2}}X^{\frac{c_{0}}{\sqrt{|G|}}}$

for any $c_{1}\geq\frac{1}{4\sqrt{17}}+\frac{\sqrt{17}}{4\log 2}\approx 1.547$
and any $c_{0}\geq\frac{\sqrt{17}}{2}\approx 2.061$. Finally, if $p=13$, we
find

$\displaystyle\\#\mathcal{F}_{k}(X;\mathbb{F}_{13}\rtimes\mathbb{F}_{13}^{\times})$
$\displaystyle\leq
e^{dp(p-1)-1}p^{3d(p-1)}\cdot(2\pi)^{2}e^{d(p-1)-1}2^{d}(p-1)^{3d}\cdot
X^{\frac{1}{12}}\cdot\left(1+\frac{\log X}{39}\right)$ $\displaystyle\leq
e^{d|G|}(2d|G|^{2})^{c_{1}d|G|^{1/2}}X^{\frac{5}{26}}$

for any $c_{1}\geq\frac{1}{2\sqrt{39}}+\frac{13}{2\sqrt{39}\cdot\log 2}\approx
1.581$ on using that $1+\frac{\log X}{39}\leq X^{1/39}$ for every $X\geq 1$;
this is consistent with any $c_{0}\geq\frac{5\sqrt{39}}{13}\approx 2.401$.

Now, again let $C=C_{G}(N)$, so that we may assume that either $|C|\geq 2p$ or
$[G:C]\leq\frac{p-1}{2}$, either of which implies that $|C|\geq\sqrt{2|G|}$.
Using Lemma 3.7, we find that

$\displaystyle\\#\mathcal{F}_{k}(X;G)$ $\displaystyle\leq
e^{d\frac{|G|}{|C|}p-1}|G|^{\frac{\log|G|}{\log 2}}p^{2d\frac{|G|}{|C|}}\cdot
2\pi\cdot
e^{d\frac{|G|}{p}}(2d|G|^{2}p^{-2})^{c_{1}d|G|^{1/2}p^{-1/2}}X^{\frac{2p_{0}}{(p_{0}-1)|C|}+\frac{c_{0}}{\sqrt{p|G|}}}$
$\displaystyle\leq
e^{d|G|}(2d|G|^{2})^{c_{1}d|G|^{1/2}}X^{\frac{c_{0}}{\sqrt{|G|}}}$

for any

$c_{1}\geq\left(\frac{1}{\sqrt{2}\cdot\log
2}+\frac{\log(2\pi/e)}{\sqrt{10}}\right)\cdot\left(1-\frac{1}{\sqrt{3}}\right)^{-1}\approx
3.040$

and any $c_{0}\geq 2\sqrt{3}+2\approx 5.464$. Here, we have exploited that we
may assume that $|G|\geq 10$ if $p\geq 5$, and that $|G|\geq 12$ if $p=3$.

This completes the treatment of Case 1), with the upshot being that we must
assume that $c_{0}\geq\frac{4}{3}(\sqrt{6}+\sqrt{3})\approx 5.575$ and
$c_{1}\geq 3.771$.

_Case 2)_ We assume that $N\simeq\mathbb{F}_{p}^{r}$ for some $r\geq 2$ and
that $|N|\geq|G|^{1/2}$. As in the proof of Theorem 1.4, by appealing to Lemma
3.3, we find

$\\#\mathcal{F}_{k}(X;G)\leq
p^{3d|G|/|N|}2^{d|G|/|N|}e^{dp|G|/|N|-d|G|/|N|-1}\cdot 2\pi\cdot
X^{\frac{2}{(p-1)|N|}}\sum_{F\in\mathcal{F}_{k}(X^{1/|N|};G/N)}\\!\\!|\mathrm{Disc}(F)|^{\frac{3}{4}-\frac{p}{p-1}}.$

We treat the inner summation via partial summation and our induction
hypothesis. For convenience, let
$C=e^{d|G/N|}(2d|G/N|^{2})^{c_{1}d|G/N|^{1/2}}$. If
$c_{0}/\sqrt{|G/N|}\leq\frac{p}{p-1}-\frac{1}{2}$, then our induction
hypothesis implies that $\\#\mathcal{F}_{k}(T;G/N)\leq
CT^{\frac{c_{0}}{\sqrt{|G/N|}}}\leq CT^{\frac{p}{p-1}-\frac{1}{2}}$ for every
$T\geq 1$, from which we find that

$\sum_{F\in\mathcal{F}_{k}(X^{1/|N|};G/N)}|\mathrm{Disc}(F)|^{\frac{3}{4}-\frac{p}{p-1}}\leq
4C\left(\frac{p}{p-1}-\frac{1}{2}\right)\cdot X^{\frac{1}{4|N|}}\leq
6CX^{\frac{1}{4|N|}}.$

On the other hand, if $c_{0}/\sqrt{|G/N|}>\frac{p}{p-1}-\frac{1}{2}$, we find

$\sum_{F\in\mathcal{F}_{k}(X^{1/|N|};G/N)}|\mathrm{Disc}(F)|^{\frac{3}{4}-\frac{p}{p-1}}\leq
4Cc_{0}X^{\frac{c_{0}}{\sqrt{|G||N|}}-\frac{p}{(p-1)|N}+\frac{3}{4|N|}}.$

Assuming that $c_{0}\leq 6$, the constant above is at most $24C$. Inserting
these estimates into our bound on $\mathcal{F}_{k}(X;G)$, we deduce that

$\\#\mathcal{F}_{k}(X;G)\leq
e^{d|G|}(2d|G|^{2})^{c_{1}d|G|^{1/2}}X^{\frac{c_{0}}{\sqrt{|G|}}}$

for any $c_{0}\geq\frac{3}{4}(2+\sqrt{2})\approx 2.560$, and with any

$c_{1}\geq\left(1+\frac{\log(48\pi/e)}{2\log 2}\right)(1+\sqrt{2})\approx
13.304$

provided that $c_{0}\leq 6$.

_Case 3)_ We assume that $N\simeq\mathbb{F}_{p}^{r}$ for some $r\geq 2$ and
that $|N|\leq|G|^{1/2}$. Let $W$ and $H$ be as in the proof of Theorem 1.4. As
there, we may assume that if $N\simeq\mathbb{F}_{2}^{3}$, then $|G|\geq 72$,
and that if $N\simeq\mathbb{F}_{2}^{2}$, then either $|G|\geq 32$ or $G\simeq
S_{4}$. Appealing to Lemma 3.7, we find that if $G\not\simeq S_{4}$ that

$\displaystyle\\#\mathcal{F}_{k}(X;G)$ $\displaystyle\leq
e^{d\frac{|G|}{|H|}p-1}|G|^{\frac{\log|G|}{\log 2}}p^{3d\frac{|G|}{|H|}}\cdot
2\pi\cdot
X^{\frac{2p_{0}}{(p_{0}-1)|H|}}\cdot\\#\mathcal{F}_{k}(X^{1/|N|};G/N)$
$\displaystyle\leq
e^{d|G|}(2d|G|^{2})^{c_{1}d|G|^{1/2}}X^{\frac{c_{0}}{\sqrt{|G|}}}$

for any $c_{0}\geq 6$ and any $c_{1}\geq 2\log\left(\frac{2\pi}{e}\right)/\log
2\approx 2.417$, where $p_{0}$ is the least prime dividing the order of $H$.

If $G\simeq S_{4}$, then we use the explicit bound

$\\#\mathcal{F}_{k}(X;S_{3})\leq
e^{6d-1}3^{6d}(2\pi)^{2}e^{2d-1}2^{4d}\cdot\frac{8}{5}\cdot
X^{\frac{17}{12}}=\frac{8}{5}(2\pi)^{2}e^{8d-2}3^{6d}2^{4d}X^{\frac{17}{12}}$

for all $X\geq 1$ proved earlier (since
$S_{3}\simeq\mathbb{F}_{3}\rtimes\mathbb{F}_{3}^{\times}$), Lemma 3.3, and
partial summation to deduce that

$\displaystyle\\#\mathcal{F}_{k}(X;S_{4})$ $\displaystyle\leq
2^{24d}e^{6d-1}\cdot 2\pi\cdot
X\cdot\sum_{F\in\mathcal{F}_{k}(X^{1/4};S_{3})}|\mathrm{Disc}(F)|^{-\frac{5}{4}}$
$\displaystyle\leq\frac{68}{5}\frac{(2\pi)^{3}}{e^{3}}e^{14d}\cdot
2^{28d}\cdot 3^{6d}\cdot X^{\frac{25}{24}}$ $\displaystyle\leq
e^{d|S_{4}|}(2d|S_{4}|^{2})^{c_{1}d|S_{4}|^{1/2}}X^{\frac{c_{0}}{\sqrt{|S_{4}|}}}$

for any $c_{0}\geq\frac{25}{\sqrt{24}}\approx 5.103$ and any $c_{1}\geq
0.612$. This completes Case 3), and the proof of Theorem 1.3 in this case. ∎

##  4\. Groups with a unique nonabelian minimal normal subgroup

In this section, by using tools developed in [Lem23], we prove Theorems
1.3–1.5 in the case that $G$ has a unique minimal normal subgroup $N$ and $N$
is nonabelian. We begin by stating an explicit bound, which also suffices to
prove Theorem 1.4.

######  Theorem 4.1.

There are constants $c_{1},c>0$ such that the following hold. Let $G$ be a
finite group with a unique minimal normal subgroup $N$, and suppose that $N$
is not abelian. Then for every number field $k$ and every $X\geq 1$, we have

$\\#\mathcal{F}_{k}(X;G)\leq
e^{d|G|}(2d|G|^{2})^{c_{1}d|G|^{1/2}}X^{\frac{c}{\sqrt{|G|}}}$

where $d=[k:\mathbb{Q}]$. In fact, we may take any $c_{1}\geq 18.5$ and any
$c\geq\frac{6935}{18\sqrt{9690}}=3.913\dots$.

Unlike the case that all of the minimal normal subgroups of $G$ are abelian,
here, we can be rather more precise about the groups under consideration. We
will use this description to provide a direct proof of Theorem 4.1, rather
than proceeding by induction as in the previous section.

######  Lemma 4.2.

Let $G$ be a finite group, and suppose that $N=T^{r}$ is the unique minimal
normal subgroup of $G$ for some nonabelian simple group $T$ and some $r\geq
1$. Then $G$ is isomorphic to a subgroup of $\mathrm{Aut}(T)\wr S_{r}$ that
acts transitively on $T^{r}$.

###### Proof.

Let $K$ denote the kernel of the map $G\to\mathrm{Aut}(N)$ given by
conjugation. Since $N$ is nonabelian, $N\not\leq K$, and hence $K=1$ by the
assumption that $N$ is the unique minimal normal subgroup of $G$. It thus
follows that $G$ is isomorphic to a subgroup of
$\mathrm{Aut}(N)\simeq\mathrm{Aut}(T)\wr S_{r}$. The claim about transitivity
follows from the fact that $N$ is minimal. ∎

Now, recall that a finite group $G$ is called _almost simple_ if it has a
unique minimal normal subgroup $N$ and $N=T$ is a nonabelian simple group, and
that the subgroup $T$ is referred to as the socle of $G$. Using ideas from
[Lem23], we first give a bound on $\\#\mathcal{F}_{k}(X;G)$ for almost simple
groups $G$. This will be the key input in the proof of Theorem 4.1.

######  Theorem 4.3.

Let $G$ be an almost simple group. Then there are constants $n=n(G)$,
$a=a(G)$, $w=w(G)$, and $\gamma=\gamma(G)$ such that for any number field $k$
and any $X\geq 1$, there holds

$\\#\mathcal{F}_{k}(X;G)\leq(2\pi)^{dn/2}(\gamma
d+1)!^{n}|G|^{dn}(2dn^{3})^{dnw}X^{\frac{a}{\sqrt{|G|}}}|\mathrm{Disc}(k)|^{-\frac{5}{4}},$

where $d:=[k:\mathbb{Q}]$. Admissible values of $n(G)$, $a(G)$, $w(G)$, and
$\gamma(G)$ are provided in Lemmas 4.9–4.16 (in particular, see Tables 1–3),
and $a(G)=4$ is admissible for every $G$.

The proof of Theorem 4.3 makes use of the classification of finite simple
groups, and in particular proceeds by an analysis of the different socle types
$T$. For the sake of organization, we carry out this proof in the immediately
subsequent subsection, before returning to the proof of Theorem 4.1.

###  4.1. Proof of Theorem 4.3

As the notation and casework involved in the proof may be distracting at first
glance, we begin by providing the essential idea of the proof. We will prove
Theorem 4.3 by instead bounding certain non-Galois extensions $F/k$, whose
normal closure has Galois group $G$. This is made clear by the following
lemma.

######  Lemma 4.4.

Let $G$ be a finite group, and let $\pi$ be a faithful and transitive
permutation representation of $G$. Let $n:=\deg\pi$. Then for any $X\geq 1$,

$\\#\mathcal{F}_{k}(X;G)\leq\\#\mathcal{F}_{n,k}(X^{\frac{n}{|G|}};\pi(G)),$

where
$\mathcal{F}_{n,k}(X;\pi(G)):=\\{F/k:[F:k]=n,\mathrm{Gal}(\widetilde{F}/k)\simeq_{\mathrm{perm}}\pi(G)\\}$,
$\widetilde{F}$ is the normal closure of $F/k$, we view
$\mathrm{Gal}(\widetilde{F}/k)$ as a permutation group via its action on the
$n$ embeddings $F\hookrightarrow\widetilde{F}$ fixing $k$, and the isomorphism
$\mathrm{Gal}(\widetilde{F}/k)\simeq_{\mathrm{perm}}\pi(G)$ is an isomorphism
of permutation groups.

###### Proof.

Let $K\in\mathcal{F}_{k}(X;G)$ and let $G_{0}$ be the stabilizer of a point in
$\pi(G)$, and note that $[G:G_{0}]=n$ by the assumption that $\pi$ is
transitive. It follows that the subfield $K^{G_{0}}=:F$ of $K$ fixed by
$G_{0}$ has degree $n$ over $k$. By the assumption that $\pi$ is faithful, we
find that $\widetilde{F}=K$, so that $K$ is determined by $F$. Moreover, we
find $\mathrm{Gal}(\widetilde{F}/k)\simeq\pi(G)$ and
$|\mathrm{Disc}(F)|\leq|\mathrm{Disc}(K)|^{\frac{n}{|G|}}\leq
X^{\frac{n}{|G|}}$. The result follows. ∎

Recent work of the author [Lem23] gives bounds on the sets
$\mathcal{F}_{n,k}(X^{\frac{n}{|G|}};\pi(G))$ in terms of the degrees of
invariants of $G$ in certain actions (where $\pi$ and $n$ are as in Lemma
4.4). We make this discussion more precise shortly, but we note at this stage
that the quantity $n(G)$ in Theorem 4.3 will always be the degree of a
faithful and transitive permutation representation $\pi$ for which we obtain
sufficiently strong bounds on
$\\#\mathcal{F}_{n,k}(X^{\frac{n}{|G|}};\pi(G))$.

By virtue of the permutation representation $\pi$ in Lemma 4.4, $G$ acts on
$\mathbb{Z}[x_{1},\dots,x_{n}]$ by $x_{i}^{g}=x_{i^{\pi(g)}}$ and its natural
extension, and we refer to an element of $f\in\mathbb{Z}[x_{1},\dots,x_{n}]$
as a $G$-invariant if it is fixed by this action. We say a $G$-invariant $f$
is homogeneous if it is homogeneous as a polynomial, and we say it is monic if
each of its non-zero coefficients is $1$. Finally, given a set
$\mathcal{I}=\\{f_{1},\dots,f_{n}\\}$ of algebraically independent, monic,
homogeneous $G$-invariants, the methods of [Lem23] give bounds on
$\mathcal{F}_{n,k}(X;\pi(G))$ in terms of the degrees of the invariants
$f_{1},\dots,f_{n}$. Combining these methods with Lemma 4.4, we obtain:

######  Lemma 4.5.

Let $G$ be an almost simple group and let $\pi$ be a faithful and transitive
permutation representation of $G$. Set $n=\deg\pi$. Suppose there is a set
$\\{f_{1},\dots,f_{n}\\}$ of $n$ algebraically independent, monic, homogeneous
$G$-invariants in $\mathbb{Z}[x_{1},\dots,x_{n}]$. Then the conclusion of
Theorem 4.3 holds with $n(G)=\deg\pi$, $w(G)=\frac{1}{n}\sum_{i=1}^{n}\deg
f_{i}$, $\gamma(G)=1$, and

$a(G)=\frac{1}{\sqrt{|G|}}\sum_{i=1}^{n}\left(\deg f_{i}-\frac{1}{2}\right).$

###### Proof.

Suppose first that $\sum_{i=1}^{n}\deg f_{i}\leq\frac{n(n+1)}{2}$. Then this
follows from [Lem23, Theorem 3.8] and Lemma 4.4. (See also the proof of
[Lem23, Lemma 5.1].) If $\sum_{i=1}^{n}\deg f_{i}>\frac{n(n+1)}{2}$, then
[Lem23, Theorem 2.19] and Lemma 4.4 yield a strictly stronger result. (See
also Lemma 4.6 below.) ∎

Now, for most almost simple groups, it follows from [Lem23, Theorem 1.10] that
there is always a set of invariants as in Lemma 4.5 satisfying $\max\\{\deg
f_{i}\\}\leq C\frac{\log|G|}{\log n}$ for some absolute constant $C$. (In
particular, this holds unless $G$ is an almost simple group of classical type
containing the coset of a graph automorphism. The quantity $\gamma(G)$ will
always be $1$ unless $G$ does contain a coset of a graph automorphism.) We
therefore find the qualitative bound
$\\#\mathcal{F}_{k}(X;G)\ll_{[k:\mathbb{Q}],G}X^{O\left(\frac{n\log|G|}{|G|\log
n}\right)}$. As a consequence of the classification of finite simple groups
(see Lemma 4.17 below), for every almost simple group $G$, there is a faithful
and transitive permutation representation of degree $n\leq\sqrt{|G|}$, which
therefore implies the qualitative bound
$\\#\mathcal{F}_{k}(X;G)\ll_{[k:\mathbb{Q}],G}X^{O\left(1/{\sqrt{|G|}}\right)}$,
which agrees with the claim in Theorem 4.3.

To make the exponent explicit, however, it is necessary to delve deeper into
the classification. For the most part, the results of [Lem23] are sufficient
to establish the bound $a(G)\leq 4$ without additional work (which suffices
for both Theorems 1.3 and 1.4), but there are a few groups that require a more
careful analysis. Before turning to this analysis, we find it convenient to
record two further results along the lines of Lemma 4.5. The first will be
useful for treating groups with particularly small degree permutation
representations, the second for treating almost simple groups containing the
coset of a graph automorphism.

######  Lemma 4.6.

Let $G$ be an almost simple group, let $\pi$ be a faithful and primitive
permutation representation of $G$, let $n=\deg\pi$, and suppose that
$\pi(G)\leq S_{n_{0}}\wr S_{d}$ for some integers $n_{0}\geq d\geq 1$ with
$n=n_{0}d$. Then the conclusion of Theorem 4.3 holds for $G$, with $n(G)=n$,
$w(G)=1/6$, $\gamma(G)=1$, and

$a(G)=\frac{dn_{0}(n_{0}+2)}{4\sqrt{|G|}}.$

###### Proof.

This follows from [Lem23, Theorem 2.19, (2.2)] and Lemma 4.4. ∎

######  Lemma 4.7.

Let $G$ be an almost simple group containing the coset of a graph
automorphism. Let $G_{0}$ be the largest subgroup of $G$ not containing the
coset of a graph automorphism, and let $\gamma:=[G:G_{0}]$. Let $\pi_{0}$ be a
faithful and transitive permutation representation of $G_{0}$ with
$\deg\pi_{0}=:n_{0}$ and let $\\{f_{1},\dots,f_{n_{0}}\\}$ be an algebraically
independent set of $G_{0}$-invariants.

Then the conclusion of Theorem 4.3 holds with $n(G)=\gamma n_{0}$,
$\gamma(G)=\gamma$,

$w(G)=\frac{1}{n_{0}}\sum_{i=1}^{n_{0}}\deg f_{i},\text{ and
}a(G)=\frac{\gamma}{\sqrt{|G|}}\sum_{i=1}^{n_{0}}\left(\deg
f_{i}-\frac{1}{2}\right).$

###### Proof.

Let $H_{0}\leq G_{0}$ be such that the coset action of $G_{0}$ on $H_{0}$ is
isomorphic to $\pi_{0}$, that is, $H_{0}$ is the stabilizer in
$\pi_{0}(G_{0})$ of a point. Let $\pi$ be the permutation representation of
$G$ corresponding to its action on the cosets of $H_{0}$, and note that
$\deg\pi=\gamma\deg\pi_{0}$ and that $\pi$ is transitive by construction.
Moreover, because $\pi_{0}$ is assumed to be faithful, it follows that
$\mathrm{Core}_{G_{0}}(H_{0}):=\bigcap_{g\in G_{0}}H_{0}^{g}$ is trivial.
Hence we also find that $\mathrm{Core}_{G}H_{0}=1$, so that $\pi$ is faithful
as well. Thus, by Lemma 4.4, we find

$\\#\mathcal{F}_{k}(X;G)\leq\\#\mathcal{F}_{n,k}(X^{\frac{n}{|G|}};\pi(G)).$

We next observe that any $\pi(G)$-extension of $k$ may be realized as a
$\pi_{0}(G_{0})$-extension of an extension of $k$ with degree $\gamma$.
Appealing to [Lem23, Theorem 3.8], [Lem23, Theorem 2.19], and [Lem23,
Proposition 2.18], we find the result. ∎

Finally, we record the classification of finite simple groups in the form that
we shall use it.

######  Lemma 4.8.

Let $T$ be a nonabelian finite simple group. Then $T$ is isomorphic to one of
the following:

  1. i)

an alternating group $A_{n}$, for some $n\geq 5$;

  2. ii)

a “classical group” of the form $\mathrm{PSL}_{m}(\mathbb{F}_{q})$,
$\mathrm{PSp}_{2m}(\mathbb{F}_{q})$, $\mathrm{PSU}_{m}(\mathbb{F}_{q})$,
$\mathrm{P\Omega}^{+}_{2m}(\mathbb{F}_{q})$,
$\mathrm{P\Omega}^{-}_{2m}(\mathbb{F}_{q})$, or
$\mathrm{P\Omega}_{2m+1}(\mathbb{F}_{q})$ for some integer $m\geq 2$ and some
prime power $q$;

  3. iii)

an “exceptional group” of the form $G_{2}(q)$, $F_{4}(q)$, $E_{6}(q)$,
$E_{7}(q)$, $E_{8}(q)$, $\mathbin{{}^{2}E_{6}}(q^{2})$,
$\mathbin{{}^{3}D_{4}}(q^{3})$, $\mathbin{{}^{2}B_{2}}(2^{2r+1})$,
$\mathbin{{}^{2}F_{4}}(2^{2r+1})$, $\mathbin{{}^{2}G_{2}}(3^{2r+1})$, or
$\mathbin{{}^{2}F_{4}}(2)^{\prime}$, where $q$ is a prime power and $r\geq 1$
is an integer; or

  4. iv)

one of the $26$ “sporadic groups” customarily denoted $\mathrm{M}_{11}$,
$\mathrm{M}_{12}$, $\mathrm{M}_{22}$, $\mathrm{M}_{23}$, $\mathrm{M}_{24}$,
$\mathrm{J}_{1}$, $\mathrm{J}_{2}$, $\mathrm{J}_{3}$, $\mathrm{HS}$,
$\mathrm{McL}$, $\mathrm{Co}_{3}$, $\mathrm{Co}_{2}$, $\mathrm{He}$,
$\mathrm{Suz}$, $\mathrm{Fi}_{22}$, $\mathrm{Ru}$, $\mathrm{Fi}_{23}$,
$\mathrm{J}_{4}$, $\mathrm{Ly}$, $\mathrm{Co}_{1}$, $\mathrm{HN}$,
$\mathrm{O^{\prime}N}$, $\mathrm{Th}$, $\mathrm{Fi}_{24}^{\prime}$,
$\mathbb{B}$, or $\mathbb{M}$.

We comment further on our conventions regarding these groups as necessary.
Additionally, beyond the statement of the classification provided in Lemma
4.8, we will also make use of the properties of these groups (notably the
computation of their outer automorphism groups), and, sometimes directly and
sometimes indirectly, their character tables and fusion maps as presented in
the ATLAS of Finite Groups [CCN+85] and its implementation in GAP [GAP24].

####  4.1.1. Alternating groups

For alternating groups, Lemma 4.6 is sufficient for our purposes.

######  Lemma 4.9.

Let $n\geq 5$, let $A_{n}$ denote the alternating group of degree $n$, and let
$G$ be an almost simple group with socle $A_{n}$. If $n\neq 6$, or $G\simeq
A_{6},S_{6}$, then Theorem 4.3 holds for $G$, with $n(G)=n$,
$w(G)=\frac{1}{2}$, $\gamma(G)=1$, and
$a(G)=\frac{n(n+2)}{4\sqrt{n!/2}}\leq\frac{35}{8\sqrt{15}}<1.130$. If $n=6$
and $G\not\simeq A_{6},S_{6}$, then Theorem 4.3 holds for $G$ with $n(G)=10$,
$w(G)=\frac{1}{6}$, $\gamma(G)=1$, and $a(G)=\frac{5}{2\sqrt{5}}<1.119$.

###### Proof.

Provided that $n\neq 6$, this follows immediately from Lemma 4.6 applied to
the standard degree $n$ permutation representation of $G$, as does the case
that $G\simeq A_{6},S_{6}$. If $n=6$ and $G\not\simeq A_{6},S_{6}$, then $G$
has a primitive permutation representation in degree $10$ by virtue of the
isomorphism $A_{6}\simeq\mathrm{PSL}_{2}(\mathbb{F}_{9})$. The lemma then
follows from Lemma 4.6 applied to that degree $10$ representation, and yields
that $a(G)=\frac{5}{2\sqrt{5}}$ is admissible for these groups. Finally, a
straightforward computation shows that the largest of these values of $a(G)$
arises for $n=5$, which yields the claim $a(G)<1.130$ in the lemma. ∎

####  4.1.2. Classical groups

For classical groups, we rely much more heavily on the results of [Lem23]. In
particular, [Lem23, Lemmas 4.8–4.17] provide bounds of the degrees of
invariants as in Lemma 4.5 depending only on the rank of the group and not on
the size of the underlying finite field. Since the minimal degree of a
classical group $G$ is typically much smaller than $\sqrt{|G|}$ (in fact,
always by a factor at least $O(q^{1/2})$, and typically much more), this
provides strong bounds on $a(G)$ for any classical group. Thus, for each
possible classical socle, we provide generic bounds leading to Theorem 4.3
(which will depend on whether or not there is a graph automorphism), and we
identify for each general socle type the almost simple group $G$ for which our
methods are weakest (i.e., yield the largest value of $a(G)$). This latter
identification sometimes requires some ad hoc computations, which we describe
as necessary. We perform these computations in Magma [BCP97]. The code used is
available at https://lemkeoliver.github.io/. Table 1 summarizes the main
results for these groups.

######  Lemma 4.10.

Let $T$ be a simple classical group, i.e. a group of the form
$\mathrm{PSL}_{m}(\mathbb{F}_{q})$, $\mathrm{PSp}_{2m}(\mathbb{F}_{q})$,
$\mathrm{PSU}_{m}(\mathbb{F}_{q})$,
$\mathrm{P\Omega}^{+}_{2m}(\mathbb{F}_{q})$,
$\mathrm{P\Omega}_{2m}^{-}(\mathbb{F}_{q})$, or
$\mathrm{P\Omega}_{2m+1}(\mathbb{F}_{q})$ for some integer $m\geq 2$ and some
prime power $q$. Let $G$ be an almost simple group with socle $T$. Then
Theorem 4.3 holds for $G$, with the values $n(G)$, $a(G)$, $w(G)$, and
$\gamma(G)$ recorded in Table 1. Moreover, we may take $a(G)=2.248$ for every
such $G$.

Socle | Conditions | $n(G)$ | $a(G)$ | $w(G)$ | $\gamma(G)$
---|---|---|---|---|---
$\mathrm{PSL}_{m}(\mathbb{F}_{q})$ | $G\leq\mathrm{P\Gamma L}_{m}(\mathbb{F}_{q})$ | $\displaystyle\frac{q^{m}-1}{q-1}$ | $1.453$ | $5m+5$ | $1$
|  $m\geq 3$, $G\not\leq\mathrm{P\Gamma L}_{m}(\mathbb{F}_{q})$ | $\displaystyle\frac{2(q^{m}-1)}{q-1}$ | $1.202$ | $5m+5$ | $2$
$\mathrm{PSp}_{2m}(\mathbb{F}_{q})$ | no graph aut. | $\displaystyle\frac{q^{2m}-1}{q-1}$ | $1.256$ | $7m+5$ | $1$
|  $m=2$, $q$ even, graph aut. | $\displaystyle\frac{2(q^{4}-1)}{q-1}$ | $2.248$ | $19$ | $2$
$\mathrm{PSU}_{m}(\mathbb{F}_{q})$ | $m=3$ | $\displaystyle q^{3}+1$ | $2.076$ | $21$ | $1$
| $m=4$ | $(q^{3}+1)(q+1)$ | $0.841$ | $25$ | $1$
|  $m\geq 5$ odd | $\displaystyle\frac{(q^{m}+1)(q^{m-1}-1)}{q^{2}-1}$ | $1.272$ | $\displaystyle\frac{7m+23}{2}$ | $1$
|  $m\geq 6$ even | $\displaystyle\frac{(q^{m}-1)(q^{m-1}+1)}{q^{2}-1}$ | $0.214$ | $\displaystyle\frac{7m+18}{2}$ | $1$
$\mathrm{P\Omega}^{+}_{2m}(\mathbb{F}_{q})$ | no graph aut. | $\displaystyle\frac{(q^{m}-1)(q^{m-1}+1)}{q-1}$ | $0.374$ | $7m+9$ | 1
|  $m=4$, graph aut. | $\displaystyle\frac{3(q^{4}-1)(q^{3}+1)}{q-1}$ | $0.647$ | $37$ | $3$
$\mathrm{P\Omega}^{-}_{2m}(\mathbb{F}_{q})$ | $m\geq 4$ | $\displaystyle\frac{(q^{m}-1)(q^{m-1}+1)}{q-1}$ | $0.409$ | $7m+15$ | $1$
$\mathrm{P\Omega}_{2m+1}(\mathbb{F}_{q})$ |  $m\geq 3$, $q$ odd | $\displaystyle\frac{q^{2m}-1}{q-1}$ | $0.197$ | $7m+16$ | $1$
Table 1. Bounds on classical almost simple groups

###### Proof.

We proceed through the different socle types in turn.

(Linear groups) Suppose $T=\mathrm{PSL}_{m}(\mathbb{F}_{q})$ for some $m\geq
2$ and some prime power $q$, with $q=8$ or $q\geq 11$ if $m=2$, and with
$q\geq 3$ if $m=3$. (We may make these assumptions since
$\mathrm{PSL}_{2}(\mathbb{F}_{4})\simeq\mathrm{PSL}_{2}(\mathbb{F}_{5})\simeq
A_{5}$, $\mathrm{PSL}_{2}(\mathbb{F}_{9})\simeq A_{6}$, and
$\mathrm{PSL}_{3}(\mathbb{F}_{2})\simeq\mathrm{PSL}_{2}(\mathbb{F}_{7})$. Let
$G_{0}:=G\cap\mathrm{P\Gamma L}_{m}(\mathbb{F}_{q})$, and note that
$\gamma(G)=[G:G_{0}]$. It follows from [Lem23, Lemma 4.8] that in the degree
$n_{0}:=\frac{q^{m}-1}{q-1}$ permutation representation of $G_{0}$ on
$\mathbb{P}^{m-1}(\mathbb{F}_{q})$, there is a set of invariants
$\\{f_{1},\dots,f_{n_{0}}\\}$ with $\deg f_{i}\leq 5m+5$ for each $i$. Thus,
it follows from Lemma 4.7 that Theorem 4.3 holds for $G$ with the claimed
values of $\gamma(G)$, $w(G)$, and $n(G)$, and with

(4.1) $a(G)=\frac{(10m+9)(q^{m}-1)\gamma(G)}{2(q-1)\sqrt{|G|}}.$

We now wish to show that we may also take $a(G)=1.453$ for every
$G\leq\mathrm{P\Gamma L}_{m}(\mathbb{F}_{q})$ and $a(G)=1.202$ for every
$G\not\leq\mathrm{P\Gamma L}_{m}(\mathbb{F}_{q})$.

Suppose that $G_{0}=G$. We first note that (4.1) is sufficient to obtain
$a(G)=1.453$ if $m\geq 5$, if $m=4$ and $q\geq 3$, if $m=3$ and $q\geq 5$, and
if $m=2$ and $q\geq 211$. We treat the finitely many remaining groups as
follows. First, for the groups with $m=3$ or $m=4$, Lemma 4.6 is sufficient
and shows that $a(G)=1.216$ is admissible for these groups. We therefore
restrict our attention to almost simple groups with socle
$\mathrm{PSL}_{2}(\mathbb{F}_{q})$ with $q\geq 7$, $q\neq 9$. Using the degree
$7$ and $11$ representations of $\mathrm{PSL}_{2}(\mathbb{F}_{7})$ and
$\mathrm{PSL}_{2}(\mathbb{F}_{11})$, respectively, we see that Lemma 4.6 is
sufficient for $q\leq 11$ and $q=16$. Next, using the invariants from [Lem23,
Lemma 4.9] in concert with Lemma 4.5, we see that if $q\geq 23$ is prime, or
if $q\geq 81$, then we may take

$a(G)=\frac{11q+125}{2\sqrt{|G|}},$

which is sufficient unless $13\leq q\leq 43$, $q=49$, or $q=64$. For the
fourteen remaining socle types, we compute a minimal set of algebraically
independent invariants in Magma [BCP97] as in the proof of [Lem23, Lemma
4.18]. For $G=\mathrm{PSL}_{2}(\mathbb{F}_{13})$, we find that the minimal set
of invariants has degrees $\\{1,2,3^{2},4^{4},5^{6}\\}$. Combined with Lemma
4.5, this shows we may take $a(G)=\frac{24}{\sqrt{273}}=1.452\dots$ in this
case (which gives the claimed value $a(G)=1.453$). For the remaining groups
$G$, we compute a minimal set of invariants for the group $\mathrm{P\Gamma
L}_{2}(\mathbb{F}_{q})$ and verify that they result in a strictly smaller
admissible value of $a(G)$.

Now suppose that $G_{0}\neq G$, i.e. that $G\not\leq\mathrm{P\Gamma
L}_{m}(\mathbb{F}_{q})$ and $\gamma(G)=2$. In this case, (4.1) is sufficient
unless $m=3$ and $q\leq 7$ or $m=4$ and $q=2$. In each of these cases, we
compute the bound from Lemma 4.6; the largest value of $a(G)$ arises for a
group of the form $G=\mathrm{PSL}_{3}(\mathbb{F}_{4}).2$ containing the coset
of a graph automorphism.

(Symplectic groups) Suppose $T=\mathrm{PSp}_{2m}(\mathbb{F}_{q})$ for some
$m\geq 2$ and some prime power $q$, with $q\geq 3$ if $m=2$. Let
$G_{0}=G\cap\mathrm{P\Gamma L}_{2m}(\mathbb{F}_{q})$, and note that
$\gamma(G)=[G:G_{0}]$. It follows from [Lem23, Lemma 4.10] that there is a set
of $n_{0}:=\frac{q^{2m}-1}{q-1}$ algebraically independent $G_{0}$-invariants
in its action on $\mathbb{P}^{2m-1}(\mathbb{F}_{q})$ with maximum degree at
most $7m+5$. Using Lemma 4.7, this yields the stated values of $\gamma(G)$,
$n(G)$, and $w(G)$, and with

(4.2) $a(G)=\frac{(14m+9)(q^{2m}-1)\gamma(G)}{2(q-1)\sqrt{|G|}}.$

First suppose that $\gamma(G)=1$. Then (4.2) implies that $a(G)=1.256$ is
admissible unless $m=2$ and $3\leq q\leq 5$ or $m=3$ and $q=2$. If
$T=\mathrm{PSp}_{6}(\mathbb{F}_{2})$, then we use Lemma 4.6 to see that the
claim holds with any $a(G)\leq\frac{455}{64\sqrt{70}}\approx 0.849$. Computing
a minimal set of invariants for the normalizers of
$\mathrm{PSp}_{4}(\mathbb{F}_{3})$ and $\mathrm{PSp}_{4}(\mathbb{F}_{4})$
inside $\mathrm{P\Gamma L}_{4}(\mathbb{F}_{3})$ and $\mathrm{P\Gamma
L}_{4}(\mathbb{F}_{4})$, we see that the claim follows for groups of these
socle types as well on using Lemma 4.5. Finally, noting that the invariants
$\\{f_{i}\\}_{i\leq n}$ provided by [Lem23, Lemma 4.10] satisfy $\deg f_{i}=i$
for $i\leq 18$, and $\deg f_{i}=19$ for $19\leq i\leq n$, we obtain using
Lemma 4.5 the bound $a(G)\leq\frac{181}{40\sqrt{13}}=1.255\dots$ for groups of
type $\mathrm{PSp}_{4}(\mathbb{F}_{5})$.

Now suppose that $\gamma(G)=2$, so that $m=2$ and $q\geq 4$ is even. Then
(4.2) is sufficient to obtain the bound $a(G)<2.248$ unless $q=4$.

(Unitary groups) Suppose $T=\mathrm{PSU}_{m}(\mathbb{F}_{q})$ for some $m\geq
3$ and some prime power $q$, with $q\geq 3$ if $m=3$. If $m\geq 5$ is odd,
then [Lem23, Lemma 4.13] implies that there is a set of algebraically
independent $G$-invariants in its degree $n(G)$ action with degrees at most
$\frac{7m+23}{2}$. Lemma 4.5 yields the stated values of $n(G)$, $w(G)$, and
$\gamma(G)$, and

$a(G)=\frac{(7m+22)(q^{m}+1)(q^{m-1}-1)}{2(q^{2}-1)\sqrt{|G|}}\leq\frac{1045}{64\sqrt{165}}<1.272,$

as claimed. Similarly, if $m\geq 6$ is even, then [Lem23, Lemma 4.12] implies
that there is a set of algebraically independent $G$-invariants with degrees
at most $\frac{7m+18}{2}$. This yields the claim with the stated values of
$n(G)$, $w(G)$, and $\gamma(G)$, and

$a(G)=\frac{(7m+17)(q^{m}-1)(q^{m-1}+1)}{2(q^{2}-1)\sqrt{|G|}}\leq\frac{4543}{768\sqrt{770}}<0.214,$

as claimed.

Now suppose that $m=3$. Using [Lem23, Lemma 4.13] and [Lem23, Corollary 4.4],
we find that there is a set of invariants of $G$ in its degree $q^{3}+1$
action with degree at most $21$. Lemma 4.5 thus shows that Theorem 4.3 holds
for $G$, with the claimed values of $n(G)$, $w(G)$, and $\gamma(G)$, and with

$a(G)=\frac{41(q^{3}+1)}{2\sqrt{|G|}}.$

This is sufficient to obtain the bound $a(G)<2.056$ unless $q\leq 11$ or
$q=17$. For groups of type $\mathrm{PSU}_{3}(\mathbb{F}_{3})$ and
$\mathrm{PSU}_{3}(\mathbb{F}_{4})$, we compute a minimal set of invariants,
which yield a strictly smaller value of $a(G)$. For $5\leq q\leq 17$, we
experimentally find that groups of type $\mathrm{PSU}_{3}(q)$ all have a base
of size $3$. Using [Lem23, Corollary 4.4], this implies that there is a set of
algebraically independent invariants of degree at most $10$. Together with
Lemma 4.5, this is sufficient provided that
$G\neq\mathrm{PSU}_{3}(\mathbb{F}_{5})$ and $G\neq
2.\mathrm{PSU}_{3}(\mathbb{F}_{5})$, with the bound for the group
$G=\mathrm{PSU}_{3}(\mathbb{F}_{8})$ being largest, namely
$a(\mathrm{PSU}_{3}(\mathbb{F}_{8}))=\frac{1083}{32\sqrt{266}}=2.075\dots$.
For the groups $\mathrm{PSU}_{3}(\mathbb{F}_{5})$ and
$\mathrm{PSU}_{3}(\mathbb{F}_{5}).2$, we exploit the fact that such groups
have a smaller degree $50$ permutation representation. We compute a minimal
set of invariants for these groups in their smaller representations, which
yields a strictly smaller value of $a(G)$.

Finally, suppose that $m=4$. Using [Lem23, Lemma 4.14], we see that there is a
set of independent set of $G$-invariants in its degree $(q^{3}+1)(q+1)$
representation with degrees at most $25$. By Lemma 4.5, we see that Theorem
4.3 holds for $G$, with the stated values of $n(G)$, $w(G)$, and $\gamma(G)$,
and with

$a(G)=\frac{49(q^{3}+1)(q+1)}{2\sqrt{|G|}}.$

This is smaller than the claimed value $0.841$ if $q\geq 4$. For groups of
type $\mathrm{PSU}_{4}(\mathbb{F}_{2})$, we find that a minimal set of
invariants has degrees $\\{1,2^{2},3^{4},4^{9},5^{11}\\}$, which leads to a
smaller value of $a(G)$. For the specific group
$T=\mathrm{PSU}_{4}(\mathbb{F}_{3})$, we find that $T$ has a base of size $4$
(and hence independent invariants of degree at most $15$), which leads to
$a(T)=\frac{1519}{216\sqrt{70}}=0.840\dots$. If
$G\leq\mathrm{Aut}(\mathrm{PSU}_{4}(\mathbb{F}_{3}))$ properly contains $T$,
then we find that $G$ has a base of size $5$. This yields a smaller value of
$a(G)$, completing the proof in this case.

(Orthogonal groups) Suppose $T=\mathrm{P\Omega}_{2m}^{+}(\mathbb{F}_{q})$ for
some $m\geq 4$ and some prime power $q$. Let $G_{0}$ be the largest subgroup
of $G$ not containing the coset of a graph automorphism. Appealing to [Lem23,
Lemma 4.15], there is an independent set of $G_{0}$-invariants in its degree
$\frac{(q^{m}-1)(q^{m-1}+1)}{q-1}$ action with degrees at most $7m+9$. Using
Lemma 4.7, we see that Theorem 4.3 holds with the stated values of $n(G)$,
$w(G)$, and $\gamma(G)$, and with

$a(G)=\frac{(14m+17)(q^{m}-1)(q^{m-1}+1)\gamma}{2(q-1)\sqrt{|G|}}.$

This expression is maximized when $q=2$, and yields the stated values of
$a(G)$.

Suppose now that $T=\mathrm{P\Omega}_{2m}^{-}(\mathbb{F}_{q})$ for some $m\geq
4$ and some prime power $q$. Then [Lem23, Lemma 4.17] implies that there is an
independent set of $G$-invariants in its degree
$\frac{(q^{m}+1)(q^{m-1}-1)}{q-1}$ action with degrees at most $7m+15$.
Appealing to Lemma 4.5, we find the claim, with

$a(G)=\frac{(14m+29)(q^{m}-1)(q^{m-1}+1)}{2(q-1)\sqrt{|G|}}\leq\frac{1275}{128\sqrt{595}}<0.409.$

Lastly, suppose that $T=\mathrm{P\Omega}_{2m+1}(\mathbb{F}_{q})$ for some
$m\geq 3$ and some odd prime power $q$. Then [Lem23, Lemma 4.16] implies that
there is a independent set of $G$-invariants in its degree
$\frac{q^{2m}-1}{q-1}$ action with degrees at most $7m+16$. The claim follows
from Lemma 4.5, with

$a(G)=\frac{(14m+31)(q^{2m}-1)}{2(q-1)\sqrt{|G|}}\leq\frac{6643}{648\sqrt{2730}}<0.197.$

This completes the proof of the lemma. ∎

####  4.1.3. Exceptional groups

We now turn to the treatment of exceptional groups. This is mostly a matter of
bookkeeping, using methods similar to those used for classical groups, but
with fewer ad hoc computations being necessary. Our treatment of the Tits
group $\mathbin{{}^{2}F_{4}}(2)^{\prime}$ may come across as somewhat ad hoc,
but previews how we treat certain sporadic groups whose minimal degree
permutation representation has degree comparatively close to $\sqrt{|G|}$. In
particular, we view our bounds for it as part of a systematic strategy. See
the discussion preceding Lemma 4.15 below.

The properties of exceptional groups (their orders and indices of parabolic
subgroups in particular) are standard, but we mention the works of Vasil’ev
[Vas96, Vas97, Vas98] as a convenient reference. Vasil’ev finds the minimal
degree permutation representations of exceptional groups, and these are
typically the representations we use in Lemma 4.5. In carrying this out, we
will make extensive use of the work of Burness, Liebeck, and Shalev [BLS09]
that gives upper bounds on the base sizes of such groups in these actions.
(Recall that a base of a permutation group is a collection of points whose
stabilizers intersect trivially.)

######  Lemma 4.11.

Let $T$ be a finite simple group of exceptional Lie type and let $G$ be an
almost simple group with socle $T$. Then Theorem 4.3 holds for $G$, with the
values of $n(G)$, $a(G)$, $w(G)$, and $\gamma(G)$ as recorded in Table 2.
Moreover, we may take $a(G)=2.374$ for every such $G$.

Socle | Conditions | $n(G)$ | $a(G)$ | $w(G)$ | $\gamma(G)$
---|---|---|---|---|---
$G_{2}(q)$ |  $q\geq 3$, no graph aut. | $\displaystyle\frac{q^{6}-1}{q-1}$ | $1.679$ | $15$ | $1$
|  $q=3^{r}$, graph aut. | $\displaystyle\frac{2(q^{6}-1)}{q-1}$ | $2.374$ | $15$ | $2$
$F_{4}(q)$ | no graph aut. | $\displaystyle\frac{(q^{12}-1)(q^{4}+1)}{q-1}$ | $0.025$ | $21$ | $1$
|  $q=2^{r}$, graph aut. | $\displaystyle\frac{2(q^{12}-1)(q^{4}+1)}{q-1}$ | $0.036$ | $21$ | $2$
$E_{6}(q)$ | no graph aut. | $\displaystyle\frac{(q^{9}-1)(q^{8}+q^{4}+1)}{q-1}$ | $8.277\cdot 10^{-6}$ | $28$ | $1$
| graph aut. | $\displaystyle\frac{2(q^{9}-1)(q^{8}+q^{4}+1)}{q-1}$ | $1.171\cdot 10^{-5}$ | $28$ | $2$
$E_{7}(q)$ | none | $\displaystyle\frac{(q^{14}-1)(q^{9}+1)(q^{5}+1)}{q-1}$ | $6.358\cdot 10^{-11}$ | $21$ | $1$
$E_{8}(q)$ | none | $\displaystyle\frac{(q^{30}-1)(q^{12}+1)(q^{10}+1)(q^{6}+1)}{q-1}$ | $2.313\cdot 10^{-19}$ | $15$ | $1$
$\mathbin{{}^{2}E_{6}}(q^{2})$ | none | $\displaystyle\frac{(q^{12}-1)(q^{6}-q^{3}+1)(q^{4}+1)}{q-1}$ | $2.080\cdot 10^{-4}$ | $15$ | $1$
$\mathbin{{}^{3}D_{4}}(q^{3})$ | none | $\displaystyle(q^{8}+q^{4}+1)(q+1)$ | $0.817$ | $15$ | $1$
$\mathbin{{}^{2}B_{2}}(q)$ |  $q=2^{2r+1}$, $r\geq 1$ | $\displaystyle q^{2}+1$ | $1.708$ | $10$ | $1$
$\mathbin{{}^{2}F_{4}}(q)$ |  $q=2^{2r+1}$, $r\geq 1$ | $\displaystyle(q^{6}+1)(q^{3}+1)(q+1)$ | $0.023$ | $10$ | $1$
$\mathbin{{}^{2}G_{2}}(q)$ |  $q=3^{2r+1}$, $r\geq 1$ | $\displaystyle q^{3}+1$ | $1.864$ | $10$ | $1$
$\mathbin{{}^{2}F_{4}}(2)^{\prime}$ | none | $1755$ | $1.872$ | $6$ | $1$
Table 2. Bounds on exceptional almost simple groups

###### Proof.

We begin with some general considerations to ease our discussion of the case
work. Suppose that a permutation group $G$ of degree $n$ has a base of size
$b$. Using [Lem23, Corollary 4.4], it follows that there is a set of
$G$-invariants with degrees at most $\frac{(b+1)(b+2)}{2}$. Using Lemma 4.5,
it follows that Theorem 4.3 holds for $G$, with $n(G)=n$,
$w(G)=\frac{(b+1)(b+2)}{2}$, $\gamma(G)=1$, and

(4.3) $a(G)=\frac{(b^{2}+3b+1)\cdot n}{2\sqrt{|G|}}.$

Similarly, suppose that a finite group $G$ admits an index $\gamma$ subgroup
$G_{0}$, where $G_{0}$ has a degree $n_{0}$ permutation representation with a
base of size $b$. Then Theorem 4.3 holds for $G$, with $n(G)=\gamma n_{0}$,
$w(G)=\frac{(b+1)(b+2)}{2}$, $\gamma(G)=\gamma$, and

(4.4) $a(G)=\frac{(b^{2}+3b+1)\cdot n_{0}\cdot\gamma}{2\sqrt{|G|}}.$

We now use these bounds to handle the different socle types in turn.

If $T=G_{2}(q)$ for some prime power $q\geq 3$, let $G_{0}$ be the largest
subgroup of $G$ not containing the coset of a graph automorphism. It follows
from [BLS09, Theorem 3] that in its degree $\frac{q^{6}-1}{q-1}$ action on the
cosets of the parabolic subgroup $P_{1}$, $G_{0}$ has a base of size at most
$4$. Evaluating the bounds (4.3) and (4.4), we obtain smaller values of $a(G)$
than claimed unless $q=3$. However, if $q=3$, then a computation reveals that
$G_{2}(3)$ has a base of size $3$. This yields the claimed values of $a(G)$.

If $T=F_{4}(q)$ for some prime power $q$, let $G_{0}$ be the largest subgroup
of $G$ not containing the coset of a graph automorphism. By [BLS09, Theorem
3], there is a base of size at most $5$ for the degree
$\frac{(q^{1}2-1)(q^{4}+1)}{q-1}$ representation of $G_{0}$ on the cosets of
the parabolic $P_{1}$. This gives the stated values, with the maximum $a(G)$
arising when $q=2$.

If $T=E_{6}(q)$ for some prime power $q$, let $G_{0}$ be the largest subgroup
of $G$ not containing the coset of a graph automorphism. By [BLS09, Theorem
3], there is a base of size at most $6$ for the degree
$\frac{(q^{9}-1)(q^{8}+q^{4}+1)}{q-1}$ action of $G_{0}$ on the parabolic
subgroup $P_{1}$. This leads to the stated values, with the maximum $a(G)$
arising from $q=2$.

If $T=E_{7}(q)$ for some prime power $q$, then by [BLS09, Theorem 3], there is
a base of size at most $5$ for the degree
$\frac{(q^{14}-1)(q^{9}+1)(q^{5}+1)}{q-1}$ action of $G$ on the cosets of the
parabolic subgroup $P_{1}$. This leads to the stated values, with the maximum
again arising from $q=2$.

If $T=E_{8}(q)$ for some prime power $q$, then by [BLS09, Theorem 3], there is
a base of size $4$ for the degree
$\frac{(q^{30}-1)(q^{12}+1)(q^{10}+1)(q^{6}+1)}{q-1}$ action of $G$ on the
cosets of the parabolic subgroup $P_{1}$. This leads to the stated values,
with the maximum again arising from $q=2$.

If $T=\mathbin{{}^{2}E_{6}}(q^{2})$ for some prime power $q$, then by [BLS09,
Theorem 3], there is a base of size $4$ for the degree
$\frac{(q^{12}-1)(q^{6}-q^{3}+1)(q^{4}+1)}{q-1}$ action of $G$ on the cosets
of the parabolic subgroup $P_{1,6}$. This leads to the stated values, with the
maximum again arising from $q=2$.

If $T=\mathbin{{}^{3}D_{4}}(q^{3})$ for some prime power $q$, then by [BLS09,
Theorem 3], there is a base of size $4$ for the degree $(q^{8}+q^{4}+1)(q+1)$
action of $G$ on the cosets of the parabolic subgroup $P_{2}$. This leads to
the stated values, with the maximum again arising from $q=2$.

If $T=\mathbin{{}^{2}B_{2}}(q)$ where $q=2^{2r+1}$ for some integer $r\geq 1$,
then by [BLS09, Theorem 3], there is a base of size $3$ in the degree
$q^{2}+1$ action of $G$ on the cosets of a parabolic subgroup. This leads to
the stated values, provided that $r\geq 2$, with the claimed $a(G)$ arising
when $r=2$ and $q=32$. For the group $\mathbin{{}^{2}B_{2}}(2^{3})$ and its
automorphism group, we compute explicitly a minimal set of invariants, which
yield strictly smaller values of $a(G)$.

If $T=\mathbin{{}^{2}F_{4}}(q)$ where $q=2^{2r+1}$ for some integer $r\geq 1$,
then by [BLS09, Theorem 3], there is a base of size $3$ for the degree
$(q^{6}+1)(q^{3}+1)(q+1)$ action of $G$ on the cosets of the parabolic
subgroup $P_{1}$. This leads to the stated values, with the maximum arising
from $q=8$.

If $T=\mathbin{{}^{2}G_{2}}(q)$ where $q=3^{2r+1}$ for some integer $r\geq 1$,
then by [BLS09, Theorem 3], there is a base of size $3$ for the degree
$q^{3}+1$ action of $G$ on the cosets of a parabolic subgroup. This leads to
the stated values, with the maximum arising from $q=27$.

Finally, if $T=\mathbin{{}^{2}F_{4}}(2)^{\prime}$ is the Tits group and
$G=\mathbin{{}^{2}F_{4}}(2)$ is its automorphism group, then by [BLS09,
Theorem 3], there is a base of size $3$ in the degree $1755$ representation of
$G$. This leads to a larger value of $a(G)$ (namely $3.933$) than we care to
allow. Instead, we verify explicitly that there is a set $\Sigma$ of four
points such that $\mathrm{Stab}_{G}\Sigma=1$ and
$\mathrm{Stab}_{G}\Sigma^{\prime}=1$ for each subset
$\Sigma^{\prime}\subset\Sigma$ of order $3$. Using [Lem23, Theorem 4.7], we
find that there is a set of independent $G$-invariants $\\{f_{i}\\}_{i\leq
1755}$ with $\deg f_{i}=i$ for $i\leq 4$, $\deg f_{i}=5$ for $5\leq i\leq
1743$, and $\deg f_{i}\leq 9$ for $1744\leq i\leq 1755$. These invariants are
also $T$-invariants, and the conclusion then follows from Lemma 4.5. ∎

####  4.1.4. Sporadic groups

We now turn to the treatment of sporadic groups; Table 3 provides a summary.
For most sporadic groups, we proceed analogously to our treatment of the
exceptional groups, relying on work of Burness, O’Brien, and Wilson [BOW10] in
place of [BLS09]. However, certain groups (namely $J_{1}$, $J_{3}$, $J_{3}.2$,
and $\mathrm{Th}$) will be treated similarly to the Tits group. Our treatment
of the Thompson group in particular requires some ideas not present in
previous sections. Thus, we compartmentalize our treatment of the different
socle types slightly more than in previous sections.

Group | $n(G)$ | $a(G)$ | $w(G)$ | $\gamma(G)$ | Group | $n(G)$ | $a(G)$ | $w(G)$ | $\gamma(G)$
---|---|---|---|---|---|---|---|---|---
$\mathrm{M}_{11}$ | $11$ | $0.402$ | $1/6$ | $1$ | $\mathrm{He}.2$ | $2\,058$ | $0.333$ | $15$ | $1$
$\mathrm{M}_{12}$ | $12$ | $0.137$ | $1/6$ | $1$ | $\mathrm{Suz}$ | $1\,782$ | $0.039$ | $15$ | $1$
$\mathrm{M}_{12}.2$ | $24$ | $0.193$ | $1/6$ | $1$ | $\mathrm{Suz}.2$ | $1\,782$ | $0.028$ | $15$ | $1$
$\mathrm{M}_{22}$ | $22$ | $0.199$ | $1/6$ | $1$ | $\mathrm{Fi}_{22}$ | $3\,510$ | $8.96\cdot 10^{-3}$ | $21$ | $1$
$\mathrm{M}_{22}.2$ | $22$ | $0.141$ | $1/6$ | $1$ | $\mathrm{Fi}_{22}.2$ | $3\,510$ | $8.50\cdot 10^{-3}$ | $28$ | $1$
$\mathrm{M}_{23}$ | $23$ | $0.046$ | $1/6$ | $1$ | $\mathrm{Ru}$ | $4\,060$ | $0.155$ | $15$ | $1$
$\mathrm{M}_{24}$ | $24$ | $9.98\cdot 10^{-3}$ | $1/6$ | $1$ | $\mathrm{Fi}_{23}$ | $31\,671$ | $3.22\cdot 10^{-4}$ | $21$ | $1$
$\mathrm{J}_{1}$ | $266$ | $2.948$ | $6$ | $1$ | $\mathrm{J}_{4}$ | $173\,067\,389$ | $0.177$ | $10$ | $1$
$\mathrm{J}_{2}$ | $100$ | $1.865$ | $15$ | $1$ | $\mathrm{Ly}$ | $8\,835\,156$ | $0.369$ | $10$ | $1$
$\mathrm{J}_{2}.2$ | $100$ | $1.319$ | $15$ | $1$ | $\mathrm{Co}_{1}$ | $98\,280$ | $9.89\cdot 10^{-4}$ | $21$ | $1$
$\mathrm{J}_{3}$ | $6\,156$ | $3.914$ | $6$ | $1$ | $\mathrm{HN}$ | $1\,140\,000$ | $0.656$ | $10$ | $1$
$\mathrm{J}_{3}.2$ | $6\,156$ | $2.768$ | $6$ | $1$ | $\mathrm{HN}.2$ | $1\,140\,000$ | $0.464$ | $10$ | $1$
$\mathrm{HS}$ | $100$ | $0.308$ | $21$ | $1$ | $\mathrm{O^{\prime}N}$ | $122\,760$ | $1.718$ | $10$ | $1$
$\mathrm{HS}.2$ | $100$ | $0.218$ | $21$ | $1$ | $\mathrm{O^{\prime}N}.2$ | $245\,520$ | $2.430$ | $10$ | $2$
$\mathrm{McL}$ | $275$ | $0.189$ | $21$ | $1$ | $\mathrm{Th}$ | $143\,127\,000$ | $2.139$ | $6$ | $1$
$\mathrm{McL}.2$ | $275$ | $0.134$ | $21$ | $1$ | $\mathrm{Fi}_{24}^{\prime}$ | $306\,936$ | $5.62\cdot 10^{-6}$ | $21$ | $1$
$\mathrm{Co}_{3}$ | $276$ | $0.011$ | $28$ | $1$ | $\mathrm{Fi}_{24}$ | $306\,936$ | $3.98\cdot 10^{-6}$ | $21$ | $1$
$\mathrm{Co}_{2}$ | $2\,300$ | $9.73\cdot 10^{-3}$ | $28$ | $1$ | $\mathbb{B}$ | $13\,571\,955\,000$ | $3.06\cdot 10^{-6}$ | $15$ | $1$
$\mathrm{He}$ | $2\,058$ | $0.471$ | $15$ | $1$ | $\mathbb{M}$ | $97\,239\,461\,142\,009\,186\,000$ | $1.03\cdot 10^{-6}$ | $10$ | $1$
Table 3. Bounds on sporadic almost simple groups

We begin with the Mathieu groups, for which Lemma 4.6 will be sufficient.

######  Lemma 4.12.

Let $T$ be one of the Mathieu groups $\mathrm{M}_{11}$, $\mathrm{M}_{12}$,
$\mathrm{M}_{22}$, $\mathrm{M}_{23}$, and $\mathrm{M}_{24}$, and let $G$ be an
almost simple group with socle $T$. Then Theorem 4.3 holds for $G$, with the
values $n(G)$, $a(G)$, $w(G)$, and $\gamma(G)$ as recorded in Table 3.
Moreover, we may take $a(G)=0.402$ for each such $G$.

###### Proof.

If $G$ is not simple, then either $T=\mathrm{M}_{12}$ and
$G=\mathrm{M}_{12}.2$ (which has a degree $24$ imprimitive permutation
representation) or $T=\mathrm{M}_{22}$ and $G=\mathrm{M}_{22}.2$ (which has a
degree $22$ primitive permutation representation). The lemma then follows from
Lemma 4.6. ∎

We now treat the socles for which the results from [BOW10] on base sizes of
sporadic groups will be sufficient.

######  Lemma 4.13.

Let $G$ be an almost simple group with sporadic socle not equal to a Mathieu
group, $\mathrm{J}_{1}$, $\mathrm{J}_{3}$, or the Thompson group
$\mathrm{Th}$. Then Theorem 4.3 holds for $G$, with the values $n(G)$, $a(G)$,
$w(G)$, and $\gamma(G)$ as recorded in Table 3. Moreover, we may take
$a(G)=2.430$ for each such $G$.

###### Proof.

The work of Burness, O’Brien, and Wilson [BOW10] mentioned earlier determines
exactly the minimal base of each such group in its minimal degree primitive
permutation representation. For all but $\mathrm{O^{\prime}N}.2$, we use this
minimal degree primitive representation in concert with (4.3) to conclude. For
$\mathrm{O^{\prime}N}.2$, we use (4.4). ∎

It therefore remains to treat almost simple groups with socle
$\mathrm{J}_{1}$, $\mathrm{J}_{3}$, or $\mathrm{Th}$. For $\mathrm{J}_{1}$ and
$\mathrm{J}_{3}$, we approach this largely computationally.

######  Lemma 4.14.

Let $G$ be $\mathrm{J}_{1}$, $\mathrm{J}_{3}$, or $\mathrm{J}_{3}.2$. Then
Theorem 4.3 holds for $G$, with the values $n(G)$, $a(G)$, $w(G)$, and
$\gamma(G)$ as recorded in Table 3. Moreover, we may take
$a(G)=\frac{6395}{18\sqrt{9690}}=3.913\dots$ for these $G$.

###### Proof.

We explicitly verify in Magma that in the minimal degree primitive permutation
representation of these groups, there is a set $\Sigma$ of $4$ points such
that $\mathrm{Stab}_{G}\Sigma=1$ and so that
$\mathrm{Stab}_{G}\Sigma^{\prime}=1$ for each subset
$\Sigma^{\prime}\subset\Sigma$ of order $3$. It follows from [Lem23, Theorem
4.7] that there is then a set of independent invariants $\\{f_{i}\\}$ with
degrees $\deg f_{i}=i$ for $i\leq 4$, $\deg f_{i}=5$ for $5\leq i\leq n-12$,
and $\deg f_{i}\leq 9$ for $n-11\leq i\leq n$. The claim now follows from
Lemma 4.5 and an easy computation. ∎

We now place this proof in somewhat greater context. If a permutation group
$G$ has a minimal base of size $b$, the best the general methods of [Lem23]
can do is to produce a set of independent $G$-invariants with degrees bounded
by $b+2$ apart from a small number of exceptions. This is achieved by [Lem23,
Theorem 4.7] precisely when there is a set $\Sigma$ of $b+1$ points such that
$\mathrm{Stab}_{G}\Sigma=1$ and so that every subset
$\Sigma^{\prime}\subset\Sigma$ of order $b$ forms a base for $G$. The groups
$\mathrm{J}_{1}$, $\mathrm{J}_{3}$, and $\mathrm{J}_{3}.2$ considered in Lemma
4.14 each have a minimal base of size $3$, which explains the role of the set
$\Sigma$ with $4$ points. Similarly, the Tits group
$\mathbin{{}^{2}F_{4}}(2)^{\prime}$ also has a minimal base of size $3$ in its
degree $1755$ representation, which explains our treatment of it in the proof
of Lemma 4.11. In fact, the Thompson group $\mathrm{Th}$ also has a minimal
base of size $3$ in its minimal degree permutation representation, but unlike
these other groups, this permutation representation is not stored in computer
algebra systems like GAP or Magma. We therefore instead provide a theoretical
argument that such a set $\Sigma$ must exist.

######  Lemma 4.15.

Let $G=\mathrm{Th}$ be the Thompson group of order
$90{\,}745{\,}943{\,}887{\,}872{\,}000=2^{15}\cdot 3^{10}\cdot 5^{3}\cdot
7^{2}\cdot 13\cdot 19\cdot 31$, in its minimal degree $n=143\,127\,000$
primitive permutation repreresentation. Then there is a subset
$\Sigma\subseteq\\{1,\dots,n\\}$ of size $4$ such that
$\mathrm{Stab}_{G}\Sigma=1$ and so that $\mathrm{Stab}_{G}\Sigma^{\prime}=1$
for each subset $\Sigma^{\prime}\subset\Sigma$ of size $3$.

###### Proof.

We proceed in a manner heavily inspired by the probabilistic methods of
[BOW10]. We consider the probability that a randomly chosen set $\Sigma$ does
not have this property. If this probability is strictly less than $1$, then
there must be such a $\Sigma$. Our goal, therefore, is to establish this
inequality.

Let $x,y,z,w$ be chosen uniformly at random from $\\{1,\dots,n\\}$, allowing
repetition. By the union bound, it follows that the probability that
$\mathrm{Stab}_{G}\\{x,y,z,w\\}\neq 1$, $\mathrm{Stab}_{G}\\{x,y,z\\}\neq 1$,
$\mathrm{Stab}_{G}\\{x,y,w\\}\neq 1$, $\mathrm{Stab}_{G}\\{x,z,w\\}\neq 1$, or
$\mathrm{Stab}_{G}\\{y,z,w\\}\neq 1$ is at most

(4.5) $\mathbf{Prob}[\mathrm{Stab}_{G}\\{x,y,z,w\\}\neq
1]+4\cdot\mathbf{Prob}[\mathrm{Stab}_{G}\\{x,y,z\\}\neq 1].$

Next, if either $\mathrm{Stab}_{G}\\{x,y,z,w\\}$ or
$\mathrm{Stab}_{G}\\{x,y,z\\}$ is nontrivial, then there must be an element of
prime order inside the stabilizer. It follows that the probability the
stabilizer is nontrivial is bounded by the expected number of elements of
prime order in the stabilizer. We therefore turn to evaluating this
expectation, beginning with $\mathrm{Stab}_{G}\\{x,y,z\\}$.

First, recording the above discussion concretely, we find that

$\displaystyle\mathbf{Prob}\left[\mathrm{Stab}_{G}\\{x,y,z\\}\neq 1\right]$
$\displaystyle=\frac{1}{n^{3}}\sum_{x,y,z\leq
n}\mathbf{1}(\mathrm{Stab}_{G}\\{x,y,z\\}\neq 1)$
$\displaystyle\leq\frac{1}{n^{3}}\sum_{x,y,z\leq
n}\sum_{\begin{subarray}{c}g\in\mathrm{Stab}_{G}\\{x,y,z\\}\\\ |g|\text{
prime}\end{subarray}}1$ $\displaystyle=\sum_{\begin{subarray}{c}g\in G\\\
|g|\text{ prime}\end{subarray}}\frac{1}{n^{3}}\\#\\{x,y,z\leq
n:\\{x,y,z\\}^{g}=\\{x,y,z\\}\\}.$

We therefore consider for each $g\in G$ of prime order, the number of
$x,y,z\leq n$ for which $\\{x,y,z\\}^{g}=\\{x,y,z\\}$. Since $g$ has prime
order, the action of $g$ on $\\{1,\dots,n\\}$ decomposes as a union of fixed
points and of cycles of length $|g|$. It follows from this that if
$\\{x,y,z\\}^{g}=\\{x,y,z\\}$ and $|g|\geq 5$, then in fact $x$, $y$, and $z$
must all be fixed by $g$. Hence we find in this case that

$\\#\\{x,y,z\leq n:\\{x,y,z\\}^{g}=\\{x,y,z\\}\\}=\mathrm{Fix}(g)^{3},$

where $\mathrm{Fix}(g)$ denotes the number of fixed points of $g$. If $|g|=3$
and $x$, $y$, and $z$ are not all fixed by $g$, then they must comprise a
$3$-cycle. The number of $3$-cycles is $\frac{1}{3}(n-\mathrm{Fix}(g))$, and
for any fixed $3$-cycle, there are $6$ possible assignments of $x,y,z$. We
conclude that if $|g|=3$, then

$\\#\\{x,y,z\leq
n:\\{x,y,z\\}^{g}=\\{x,y,z\\}\\}=\mathrm{Fix}(g)^{3}+2(n-\mathrm{Fix}(g)).$

Finally, if $|g|=2$ and not all of $x$, $y$, and $z$ are fixed by $g$, then
two of them must comprise a $2$-cycle, and the third must either be fixed or
equal to one of the other two. If $|g|=2$, then there are
$\frac{1}{2}(n-\mathrm{Fix}(g))$ $2$-cycles, and we find that

$\\#\\{x,y,z\leq
n:\\{x,y,z\\}^{g}=\\{x,y,z\\}\\}=\mathrm{Fix}(g)^{3}+3\cdot\mathrm{Fix}(g)\cdot(n-\mathrm{Fix}(g))+3\cdot(n-\mathrm{Fix}(g)).$

Putting this all together, we therefore find

(4.6) $\displaystyle\mathbf{Prob}\left[\mathrm{Stab}_{G}\\{x,y,z\\}\neq
1\right]$ $\displaystyle\quad\quad\quad\quad\leq\sum_{\begin{subarray}{c}g\in
G\\\ |g|\text{
prime}\end{subarray}}\frac{\mathrm{Fix}(g)^{3}}{n^{3}}+\sum_{\begin{subarray}{c}g\in
G\\\
|g|=3\end{subarray}}\frac{2(n-\mathrm{Fix}(g))}{n^{3}}+\sum_{\begin{subarray}{c}g\in
G\\\
|g|=2\end{subarray}}\frac{3\cdot(\mathrm{Fix}(g)+1)\cdot(n-\mathrm{Fix}(g))}{n^{3}},$

which is an expression that holds for any permutation group $G$ of any degree
$n$. We now specialize to the specific case of interest to us, where $G$ is
the Thompson group acting in degree $n=143\,127\,000$. Using the ATLAS of
Finite Groups [CCN+85] and its implementation in GAP [GAP24], we find there
are $10$ conjugacy classes of elements of prime order, with labels 2A, 3A, 3B,
3C, 5A, 7A, 13A, 19A, 31A, and 31B. We also find that
$\mathrm{Fix}(\mathrm{2A})=10\,200$, $\mathrm{Fix}(\mathrm{3A})=3\,510$,
$\mathrm{Fix}(\mathrm{3B})=243$, $\mathrm{Fix}(\mathrm{3C})=540$,
$\mathrm{Fix}(\mathrm{7A})=9$, $\mathrm{Fix}(\mathrm{13A})=3$, and
$\mathrm{Fix}(\mathrm{5A})=\mathrm{Fix}(\mathrm{19A})=\mathrm{Fix}(\mathrm{31A})=\mathrm{Fix}(\mathrm{31B})=0$,
where we have written $\mathrm{Fix}(\mathrm{2A})$, for example, for the value
of $\mathrm{Fix}(g)$ for any $g$ in the conjugacy class 2A. The ATLAS also
provides the size of each conjugacy class, and using this, we compute using
(4.6) that

$\mathbf{Prob}\left[\mathrm{Stab}_{G}\\{x,y,z\\}\neq
1\right]\leq\frac{419448082}{207981421875}\approx 0.002.$

We now treat $\mathbf{Prob}[\mathrm{Stab}_{G}\\{x,y,z,w\\}\neq 1]$ similarly.
As above, we find

$\displaystyle\mathbf{Prob}\left[\mathrm{Stab}_{G}\\{x,y,z,w\\}\neq 1\right]$
$\displaystyle\leq\sum_{\begin{subarray}{c}g\in G\\\ |g|\text{
prime}\end{subarray}}\frac{1}{n^{4}}\\#\\{x,y,z,w\leq
n:\\{x,y,z,w\\}^{g}=\\{x,y,z,w\\}\\}.$

If $|g|\geq 5$, then

$\\#\\{x,y,z,w\leq n:\\{x,y,z,w\\}^{g}=\\{x,y,z,w\\}\\}=\mathrm{Fix}(g)^{4}.$

If $g=3$ and $\\{x,y,z,w\\}^{g}=\\{x,y,z,w\\}$ and not all points are fixed,
then either $\\{x,y,z,w\\}$ consists of a $3$-cycle and a fixed point or a
$3$-cycle with a doubled point. We thus find

$\\#\\{x,y,z,w\leq
n:\\{x,y,z,w\\}^{g}=\\{x,y,z,w\\}\\}=\mathrm{Fix}(g)^{4}+(8\cdot\mathrm{Fix}(g)+12)\cdot(n-\mathrm{Fix}(g)).$

If $g=2$ and $\\{x,y,z,w\\}^{g}=\\{x,y,z,w\\}$ and not all points are fixed,
then $\\{x,y,z,w\\}$ must form: two disjoint $2$-cycles; a $2$-cycle plus two
fixed points (possibly the same); a $2$-cycle with a double point plus a fixed
point; a $2$-cycle with a triple point; or a $2$-cycle with two double points.
Therefore, in this case, we find

$\displaystyle\\#\\{x,y,z,w\leq n$
$\displaystyle:\\{x,y,z,w\\}^{g}=\\{x,y,z,w\\}\\}$
$\displaystyle=\mathrm{Fix}(g)^{4}+3\cdot(n-\mathrm{Fix}(g))^{2}+(12\mathrm{Fix}(g)^{2}+12\mathrm{Fix}(g)+1)\cdot(n-\mathrm{Fix}(g)).$

All told, we conclude that

(4.7) $\displaystyle\mathbf{Prob}\big{[}\mathrm{Stab}_{G}\\{x,y,z,w\\}\neq
1\big{]}$ $\displaystyle\quad\quad\leq\sum_{\begin{subarray}{c}g\in G\\\
|g|\text{
prime}\end{subarray}}\frac{\mathrm{Fix}(g)^{4}}{n^{4}}+\sum_{\begin{subarray}{c}g\in
G\\\
|g|=3\end{subarray}}\frac{(8\cdot\mathrm{Fix}(g)+12)(n-\mathrm{Fix}(g))}{n^{4}}$
$\displaystyle\quad\quad\quad+\sum_{\begin{subarray}{c}g\in G\\\
|g|=2\end{subarray}}\frac{3(n-\mathrm{Fix}(g))^{2}+(12\cdot\mathrm{Fix}(g)^{2}+12\cdot\mathrm{Fix}(g)+1)(n-\mathrm{Fix}(g))}{n^{4}},$

which, as with (4.6), is an expression that holds for any permutation group
$G$ of degree $n$. Evaluating it for the Thompson group, we find that

$\mathbf{Prob}\big{[}\mathrm{Stab}_{G}\\{x,y,z,w\\}\neq
1\big{]}\leq\frac{2992265015279081}{5090286441648234375000}\approx 5.87\cdot
10^{-7}.$

Therefore, using (4.5), we conclude that the probability that a random set
$\Sigma$ of four points does not satisfy the conclusion of the lemma is at
most $0.01$. In particular, the conclusion of the lemma is satisfied for at
least $99\%$ of the possible sets $\Sigma$, and thus for at least one. ∎

######  Lemma 4.16.

Let $G$ be the Thompson group $\mathrm{Th}$. Then Theorem 4.3 holds for $G$,
with the values $n(G)$, $a(G)$, $w(G)$, and $\gamma(G)$ recorded in Table 3.
In particular, we may take $a(G)=2.139$.

###### Proof.

Using [Lem23, Theorem 4.7] together with Lemma 4.15, we find there is a set of
invariants $\\{f_{i}\\}_{i\leq n}$ with degrees $\deg f_{i}=i$ for $i\leq 4$,
$\deg f_{i}=5$ for $5\leq i\leq n-12$, and $\deg f_{i}\leq 9$ for $n-11\leq
i\leq n$. The result then follows from Lemma 4.5. ∎

####  4.1.5. Proof of Theorem 4.3

As every possible almost simple group has been considered in Lemmas 4.9–4.16,
the proofs of these lemmas comprise the proof of Theorem 4.3. The claim that
$a(G)=4$ is admissible follows by an examination of these bounds, with the
largest value arising from $G=\mathrm{J}_{3}$. Evaluating our bound on
$\mathrm{J}_{3}$ exactly, we see that in fact any
$a(G)\geq\frac{6935}{18\sqrt{9690}}=3.913\dots$ will be admissible.

###  4.2. Proof of Theorem 4.1

We now return to Theorem 4.1. Our first task is to convert the bounds provided
by Theorem 4.3 into the form required by Theorem 4.1. This conversion is
principally carried out in the following two straightforward lemmas. The first
makes explicit the fact that each almost simple group $G$ has a permutation
representation of degree at most $\sqrt{|G|}$.

######  Lemma 4.17.

Let $G$ be an almost simple group, and let $n(G)$ be as in Theorem 4.3. Let
$\alpha=\frac{171}{2\sqrt{9690}}=0.868\dots$. Then $n(G)\leq\alpha\sqrt{|G|}$.
This value of $\alpha$ is sharp if $G=\mathrm{J}_{3}$.

###### Proof.

We verify the claim directly in Magma for the sporadic groups, and for
exceptional and classical groups with $q<100$ and small rank
($\mathrm{PSL}_{m}$, $\mathrm{PSp}_{2m}$, $\mathrm{PSU}_{m}$,
$\mathrm{P\Omega}_{2m+1}$, and $\mathrm{P\Omega}_{2m}^{\pm}$ with $m\leq 10$).
For groups of Lie type with $q>100$ or those classical groups with $m\geq 11$,
it is straightforward to verify the claim. ∎

We will also make use of the following closely related result.

######  Lemma 4.18.

Let $G$ be an almost simple group, and let $n(G)$ and $w(G)$ be as in Theorem
4.3. Let $\beta=\frac{49\sqrt{42}}{42}=7.560\dots$. Then
$n(G)w(G)\leq\beta\sqrt{|G|}$. This value of $\beta$ is sharp if
$G=\mathrm{PSU}_{3}(\mathbb{F}_{3})$.

###### Proof.

This is directly analogous to the proof of Lemma 4.17. ∎

###### Proof of Theorem 4.1.

Let $N=T^{r}$ be the unique minimal normal subgroup of $G$, where $T$ is a
finite simple group and $r\geq 1$ is an integer. First suppose that $G$ is
almost simple, i.e. that $r=1$. Appealing to Theorem 4.3, we find that

$\\#\mathcal{F}_{k}(X;G)\leq(2\pi)^{dn/2}(\gamma
d+1)!^{n}|G|^{dn}(2dn^{3})^{dnw}X^{\frac{a}{\sqrt{|G|}}},$

where $n=n(G)$, $a=a(G)$, $w=w(G)$, and $\gamma=\gamma(G)$ are as in Theorem
4.3. We begin by noting that since $|G|\geq 60$,

$|G|^{dn}\leq e^{\alpha d|G|^{1/2}\log|G|}<e^{d|G|},$

where $\alpha$ is as in Lemma 4.17. Next, since $\gamma\leq 3$ in all cases,
we observe that $(\gamma d+1)!\leq(6d)^{6d}$. From this and the fact that
$\alpha<1$, we find that

$(2\pi)^{dn/2}(\gamma d+1)!^{n}(2dn^{3})^{dnw}\leq 2^{((\log 2\pi+6\log
6)\cdot\alpha+\beta)\cdot d|G|^{1/2}}d^{(6\alpha+\beta)\cdot
d|G|^{1/2}}(|G|^{2})^{\frac{3}{4}\beta\cdot d|G|^{1/2}},$

where $\beta$ is as in Lemma 4.18. The statement of the theorem therefore
holds for $G$, for any $c_{1}\geq(\log 2\pi+6\log 6)\cdot\alpha+\beta\approx
18.495$.

Now, suppose $r\geq 2$. View $G$ as a subgroup of $\mathrm{Aut}(T)\wr S_{r}$,
let $\pi\colon\mathrm{Aut}(T)\wr S_{r}\to S_{r}$ be the natural quotient map,
let $G_{r}=\pi(G)$, and let $H_{1}=G\cap\pi^{-1}(\mathrm{Stab}_{S_{r}}1)$.
Observe that $[G:H_{1}]=r$. Let
$H_{2}=G\cap((1\times\mathrm{Aut}(T)^{r-1})\rtimes\pi(H_{1}))$. Then $H_{2}$
is a normal subgroup of $H_{1}$, and $H_{1}/H_{2}\simeq G_{0}$ for some almost
simple group $G_{0}$ with socle $T$. Moreover, since $H_{2}$ does not contain
$N=T^{r}$, the core of $H_{2}$ in $G$ is trivial.

If $K\in\mathcal{F}_{k}(X;G)$, it follows that $[K^{H_{1}}:k]=r$ and that
$K^{H_{2}}/K^{H_{1}}$ is a Galois $G_{0}$-extension. Moreover, since $H_{2}$
is core-free, the normal closure of the extension $K^{H_{2}}$ over $k$ will be
$K$. It thus follows that

$\\#\mathcal{F}_{k}(X;G)\leq\sum_{F\in\mathcal{F}_{r,k}(X^{r/|G|};G_{r})}\\#\mathcal{F}_{F}(X^{\frac{r|G_{0}|}{|G|}};G_{0}).$

Since $G_{0}$ is almost simple, we may appeal to the $r=1$ case of the the
theorem to conclude that

$\\#\mathcal{F}_{F}(X^{\frac{r|G_{0}|}{|G|}};G_{0})\leq
e^{dr|G_{0}|}(2dr|G_{0}|^{2})^{c_{1}dr|G_{0}|^{1/2}}X^{\frac{cr\sqrt{|G_{0}|}}{|G|}},$

while using [Lem23, Theorem 2.19], we find

$\\#\mathcal{F}_{r,k}(X^{\frac{r}{|G|}};G_{r})\leq(2\pi)^{d(r-1)/2}(d+1)!^{r-1}r^{\frac{d(5r-2)}{4}+1}X^{\frac{r(r+2)}{4|G|}}.$

For convenience, we collect the powers of $X^{1/\sqrt{|G|}}$ and the constants
separately. We first note that $|G|\geq r|G_{0}|T^{r-1}$, so that

$\displaystyle X^{\frac{cr\sqrt{|G_{0}|}}{|G|}+\frac{r(r+2)}{4|G|}}$
$\displaystyle=(X^{\frac{1}{\sqrt{|G|}}})^{\frac{cr\sqrt{|G_{0}|}}{\sqrt{|G|}}+\frac{r(r+2)}{4\sqrt{|G|}}}$
$\displaystyle\leq(X^{\frac{1}{\sqrt{|G|}}})^{\frac{c\sqrt{r}}{|T|^{(r-1)/2}}+\frac{(r+2)\sqrt{r}}{4|T|^{r/2}}}$
$\displaystyle\leq(X^{\frac{1}{\sqrt{|G|}}})^{\frac{c\sqrt{2}}{\sqrt{60}}+\frac{\sqrt{2}}{60}}$
$\displaystyle\leq X^{\frac{c}{\sqrt{|G|}}}$

for any
$c\geq\frac{\sqrt{2}}{60}\left(1-\frac{\sqrt{2}}{{\sqrt{60}}}\right)^{-1}=0.0288\dots$.

For the constants, we note first that
$r\sqrt{|G_{0}|}\leq\frac{1}{\sqrt{30}}\sqrt{|G|}$, which implies that

$e^{dr|G_{0}|}(2dr|G_{0}|^{2})^{c_{1}dr|G_{0}|^{1/2}}\leq
e^{d|G|}(2d|G|^{2})^{\frac{c_{1}}{\sqrt{30}}d|G|^{1/2}},$

and on also using $r\leq\frac{\sqrt{2}}{60}\sqrt{|G|}$, that

$(2\pi)^{d(r-1)/2}(d+1)!^{r-1}r^{\frac{d(5r-2)}{4}+1}\leq
2^{d|G|^{1/2}\cdot\left(\frac{\log
2\pi}{2}\cdot\frac{\sqrt{2}}{60}+\frac{\sqrt{2}}{30}-\frac{\sqrt{2}}{40\log
2}\cdot\log\left(\frac{60}{\sqrt{2}}\right)\right)}d^{d|G|^{1/2}\frac{\sqrt{2}}{30}}(|G|^{2})^{d|G|^{1/2}\frac{\sqrt{2}}{160}}.$

Pulling this together, we find that

$\\#\mathcal{F}_{k}(X;G)\leq
e^{d|G|}(2d|G|^{2})^{c_{1}d|G|^{1/2}}X^{\frac{c}{\sqrt{|G|}}}$

provided that
$c_{1}\geq\frac{\sqrt{2}}{30}\left(1-\frac{1}{\sqrt{30}}\right)^{-1}=0.057\dots$.
The limiting bounds therefore arise from the almost simple case, and the
theorem follows. ∎

###  4.3. Proofs of Theorems 1.3–1.5

We observe that Theorem 4.1 completes the proofs of Theorem 1.3 and Theorem
1.4, when combined with Lemma 2.4 and the previous work on groups all of whose
minimal normal subgroups are abelian. Moreover, an analysis of the proof of
Theorem 4.1 (particularly that the worst-case is provided by almost simple
groups) and the bounds provided by Lemmas 4.9–4.16 (or Tables 1–3) implies
that to prove Theorem 1.5, we must only prove that it holds for the group
$G=\mathrm{J}_{3}$. We therefore have the following lemma.

######  Lemma 4.19.

Let $G=\mathrm{J}_{3}$ and let $c_{0}=\frac{863441}{2880\sqrt{9690}}\approx
3.045$. Then for any number field $k$, any $X\geq 1$, and any $\epsilon>0$, we
have

$\\#\mathcal{F}_{k}(X;G)\ll_{k,\epsilon}X^{\frac{c_{0}}{\sqrt{|G|}}+\epsilon}.$

###### Proof.

As in the proof of Lemma 4.14, we find that in its degree $n:=6156$ primitive
permutation representation, there is a set of independent $G$-invariants
$\\{f_{i}\\}$ with degrees $\deg f_{i}=i$ for $i\leq 4$, $\deg f_{i}=5$ for
$5\leq i\leq 6144$, and $\deg f_{i}\leq 9$ for $6145\leq i\leq 6156$. We also
compute that the index of $G$ (in the sense of Malle) is $3040$. Using [Lem23,
Theorem 3.16], we then see that

$\\#\mathcal{F}_{6156,k}(X;\mathrm{J}_{3})\ll_{k,\epsilon}X^{\frac{863441}{246240}+\epsilon},$

where the bound is on the number of degree $6156$ $\mathrm{J}_{3}$-extensions.
Since $|\mathrm{J}_{3}|=50\,232\,960=2^{7}\cdot 3^{5}\cdot 5\cdot 17\cdot 19$,
using Lemma 4.4, we therefore find

$\\#\mathcal{F}_{k}(X;\mathrm{J}_{3})\ll_{k,\epsilon}X^{\frac{863441}{2009318400}+\epsilon}.$

This is exactly the statement of the lemma. ∎

This completes the proof of Theorem 1.5.

###  4.4. An asymptotic improvement to the shape of Theorem 4.1

Finally, we note that while the sporadic groups (in particular
$\mathrm{J}_{3}$) are the bottleneck in computing the explicit power of $X$
provided by Theorem 4.1, they _cannot_ be the bottleneck in the asymptotic
shape of the exponent, since there are only finitely many such almost simple
groups (which the proof shows dominate the bounds). We make this clear in the
following theorem, which we prove in a soft form to make the ideas clearer and
to avoid the need for extensive case work.

######  Theorem 4.20.

There is a constant $c>0$ such that the following holds. Let $G$ be a finite
group with a unique minimal normal subgroup $N$, and suppose that $N$ is not
abelian. Then for any number field $k$ and any $X\geq 1$, there holds

$\\#\mathcal{F}_{k}(X;G)\ll_{k,G}X^{\frac{c}{|G|^{4/7}}}.$

###### Proof.

We suppose first that $G$ is almost simple. For each possible group of Lie
type (regarding the prime power $q$ as a parameter), we compute the least
exponent $\delta>0$ such that

(4.8) $n(G)\cdot w(G)\ll|G|^{\delta}$

as $q\to\infty$, where $n(G)$ and $w(G)$ are as in Theorem 4.3. It is an
exercise to see that the largest such $\delta$ arises from groups of the form
$\mathbin{{}^{2}G_{2}}(q)$. In particular, (4.8) holds with
$\delta=\frac{3}{7}$ for every almost simple group of Lie type, and, since
every $\delta>0$ is admissible for the alternating groups $A_{n}$ as
$n\to\infty$, hence also for every almost simple group. Using Lemma 4.4, Lemma
4.5, and Lemma 4.7 as above, we conclude that there is some constant $C>0$
such that

$\\#\mathcal{F}_{k}(X;G)\ll_{k,G}X^{\frac{C}{|G|^{1-\delta}}}\ll
X^{\frac{C}{|G|^{4/7}}}.$

We now suppose that $N=T^{r}$ for some $r\geq 2$ and some simple group $T$.
Letting $\delta$ be admissible in (4.8) for almost simple groups with socle
$T$, we see as in the proof of Theorem 4.1 that

$\\#\mathcal{F}_{k}(X;G)\ll_{k,G}X^{\frac{r(r+2)}{4|G|}+\frac{r|G_{0}|^{\delta}}{|G|}}=X^{\frac{1}{|G|^{1-\delta}}\left(\frac{r(r+2)}{4|G|^{\delta}}+\frac{r|G_{0}|^{\delta}}{|G|^{\delta}}\right)}\ll
X^{\frac{C^{\prime}}{|G|^{1-\delta}}}\ll X^{\frac{C^{\prime}}{|G|^{4/7}}}$

for some $C^{\prime}>0$. Hence, taking $c=\max\\{C,C^{\prime}\\}$, the result
follows. ∎

##  5\. The proof of Theorems 1.1 and 1.2

In this section, we prove Theorems 1.1 and 1.2. In carrying this out, we find
it convenient to first recall some useful results.

###  5.1. Bounds on discriminants and the number of finite groups

To approach Theorem 1.1, we will make use of the following consequence of work
of Odlyzko [Odl76], that we state in a slightly simplified form. This will
limit the groups $G$ contributing nontrivially to
$\\#\mathcal{F}_{k}^{\mathrm{Gal}}(X)$ and
$\\#\mathcal{F}_{k}^{\mathrm{nc}}(X)$.

######  Lemma 5.1.

There is an absolute constant $C$ such that for any number field $k$ and any
extension $K/k$, we have $dn\leq\frac{1}{3}\log|\mathrm{Disc}(K)|+C$, where
$d=[k:\mathbb{Q}]$ and $n=[K:k]$.

###### Proof.

By [Odl76, Theorem 1], we have $|\mathrm{Disc}(K)|\geq 21^{dn}$ for $dn$
sufficiently large. Since $\log 21=3.044\dots>3$, this implies the result for
$dn$ sufficiently large, say $dn\geq N_{0}$. The constant $C$ is chosen so
that the statement remains true for the finitely many values of $dn$ that are
less than $N_{0}$. For example, $C=N_{0}$ is admissible. ∎

######  Remark.

Odlyzko’s theorem is stated as a lower bound on $|\mathrm{Disc}(K)|$ in terms
of the degree $[K:\mathbb{Q}]$, but for us it is convenient to state the
result in terms of the two parameters $d$ and $n$, even if this is somewhat
artificial.

As a consequence of Lemma 5.1, the groups $G$ contributing to
$\mathcal{F}_{k}^{\mathrm{Gal}}(X)$ all satisfy $d|G|\leq\frac{1}{3}\log X+C$.
This puts a bound on how complicated the individual groups appearing may be,
but to prove Theorem 1.1 and 1.2, we shall also need the following result of
Holt [Hol89] that provides an upper bound on the number of groups of bounded
order.

######  Lemma 5.2.

For any $N\geq 1$, the number of isomorphism classes of finite groups $G$ with
$|G|\leq N$ is bounded above by $N^{\frac{(\log N)^{2}}{6(\log
2)^{2}}+\frac{\log N}{\log 2}}$.

###### Proof.

This is [Hol89, Theorem 2], which we note relies on the classification of
finite simple groups. ∎

In particular, combining Lemmas 5.1 and 5.2, we see that the number of groups
$G$ contributing nontrivially to $\mathcal{F}_{k}^{\mathrm{Gal}}(X)$ is
$\leq\exp(O((\log\log X)^{3}))=X^{o(1)}$. This is a sufficiently small number
that there will be essentially no difficulty in adding together the
contributions from the different groups $G$. We now make this explicit by
proving Theorems 1.1 and 1.2.

###  5.2. Proof of Theorems 1.1 and 1.2

We begin with an easy lemma that will be used to control the discrepancy
between $\\#\mathcal{F}_{k}^{\mathrm{nc}}(X)$ and
$\\#\mathcal{F}_{k}^{\mathrm{Gal}}(X)$.

######  Lemma 5.3.

For any finite group $G$, let $\mathrm{CoreFree}(G):=\\#\\{H\leq G:\cap_{g\in
G}H^{g}=1\\}$ denote the number of core-free subgroups $H\leq G$. Then
$\mathrm{CoreFree}(G)\leq\exp\left(\frac{(\log|G|)^{2}}{\log 2}\right)$.

###### Proof.

Trivially, we may bound $\mathrm{CoreFree}(G)$ by the number of subgroups
$H\leq G$ (without regard to whether they are core-free). Any subgroup $H\leq
G$ is generated by its Sylow subgroups $H_{p}$. Each Sylow subgroup $H_{p}$ is
generated by its center $Z(H_{p})$ and generators of the quotient
$H_{p}/Z(H_{p})$, from which follows that $H_{p}$ is generated by at most
$v_{p}(|H_{p}|)$ elements. Hence, any subgroup $H\leq G$ is generated by at
most $\Omega(|G|)$ elements, where $\Omega(|G|)$ is the number of prime
divisors of $|G|$, counted with multiplicity.

From this, we find

$\mathrm{CoreFree}(G)\leq|G|^{\Omega(|G|)}\leq\exp\left(\frac{(\log|G|)^{2}}{\log
2}\right),$

as claimed. ∎

###### Proof of Theorem 1.2.

We first observe that for any $X\geq e^{e}$ (so that $\log\log X\geq 1$), we
have by Lemma 5.1

$\displaystyle\\#\mathcal{F}_{k}^{\mathrm{nc}}(X)$
$\displaystyle=\sum_{|G|\leq
4}\\#\mathcal{F}_{k}(X;G)+\sum_{5\leq|G|\leq\frac{1}{3d}\log
X+\frac{C}{d}}\\#\mathcal{F}_{k}(X;G)\cdot\mathrm{CoreFree}(G)$
$\displaystyle=\sum_{|G|\leq
4}\\#\mathcal{F}_{k}(X;G)+\sum_{5184\leq|G|\leq\frac{1}{3d}\log
X+\frac{C}{d}}\\#\mathcal{F}_{k}(X;G)\cdot\mathrm{CoreFree}(G)+O_{k,\epsilon}(X^{\frac{3}{8}+\epsilon}),$

where we have invoked [EV06, Proposition 1.3] to obtain the term
$O_{k,\epsilon}(X^{\frac{3}{8}+\epsilon})$.

Note that $\frac{6}{\sqrt{n}}\leq\frac{1}{12}$ for $n\geq 5184$. For any $G$
with $|G|\leq\frac{1}{3d}\log X+\frac{C}{d}$, Lemma 5.3 shows that
$\mathrm{CoreFree}(G)\leq\exp(O((\log\log X)^{2}))$, while Lemma 5.2 shows
that the number of groups $G$ with order $5184\leq|G|\leq\frac{1}{3d}\log
X+\frac{C}{d}$ is $\exp(O((\log\log X)^{3}))$, where all implied constants are
absolute. Additionally, by Theorem 1.3, for groups $G$ such that
$5184\leq|G|\leq\frac{1}{3d}\log X+\frac{C}{d}$, we find

$\\#\mathcal{F}_{k}(X;G)\ll X^{\frac{1}{3}}e^{O((\log X)^{1/2}(\log\log
X))}X^{\frac{6}{\sqrt{n}}}\ll_{\epsilon}X^{\frac{5}{12}+\epsilon}.$

All told, we find

$\sum_{5184\leq|G|\leq\frac{1}{3d}\log
X+\frac{C}{d}}\\#\mathcal{F}_{k}(X;G)\cdot\mathrm{CoreFree}(G)\ll_{\epsilon}X^{\frac{5}{12}+\epsilon}$

for any $\epsilon>0$, and therefore also

$\\#\mathcal{F}_{k}^{\mathrm{nc}}(X)=\sum_{|G|\leq
4}\\#\mathcal{F}_{k}(X;G)+O_{k,\epsilon}(X^{\frac{5}{12}+\epsilon}).$

If $k\neq\mathbb{Q}$, then for $G=C_{3}$, $C_{4}$, and $C_{2}\times C_{2}$,
the bound $\\#\mathcal{F}_{k}(X;G)\ll_{k,\epsilon}X^{\frac{1}{2}+\epsilon}$
follows from work of Wright [Wri89]. We thus find

$\\#\mathcal{F}_{k}^{\mathrm{nc}}(X)=\\#\mathcal{F}_{k}(X;C_{2})+O_{k,\epsilon}(X^{\frac{1}{2}+\epsilon}).$

The result in this case now follows from [MT23, Theorem 2].

If $k=\mathbb{Q}$, then we have the stronger asymptotic

$\\#\mathcal{F}_{\mathbb{Q}}(X;C_{2})=\frac{6}{\pi^{2}}X+O(X^{1/2}\exp(-c\cdot(\log
X)^{3/5}(\log\log X)^{-1/5}))$

for some $c>0$, as follows from known zero-free regions for $\zeta(s)$ and
$L(s,\chi_{4})$ and standard techniques; in fact, using recent work of Khale
[Kha24], it is possible to provide an explcit value of $c$. Moreover, it
follows from [FLN18, Theorem 1.7] that there is some $\delta>0$ such that

$\\#\mathcal{F}_{\mathbb{Q}}(X;C_{3})+\\#\mathcal{F}_{\mathbb{Q}}(X;C_{4})+\\#\mathcal{F}_{\mathbb{Q}}(X;C_{2}\times
C_{2})=P_{2}(\log X)\cdot X^{\frac{1}{2}}+O(X^{\frac{1}{2}-\delta}),$

where $P_{2}(\log X)$ is an explicitly computable polynomial of degree $2$ in
$\log X$, completing the proof of the theorem when $k=\mathbb{Q}$. ∎

###### Proof of Theorem 1.1.

This follows mutatis mutandis from the proof above by removing the weight
$\mathrm{CoreFree}(G)$. ∎

###  5.3. Variants

For most variations of Theorem 1.1 incorporating a weighting that decays
(e.g., weighing fields inversely to the size of their automorphism group, or
looking at fields only up to isomorphism), or that is
$O_{\epsilon}(X^{\epsilon})$ uniformly for $|G|\leq\frac{1}{3d}\log X+C$, the
proof of Theorem 1.1 shows that to obtain an asymptotic formula, it suffices
to understand the weighting only for $G=C_{2}$ and to modify the asymptotic
accordingly.

##  6\. Bounds without the classification of finite simple groups

In this final section, we provide a proof of the bound in Theorem 1.7, which
does not rely on the classification of finite simple groups. Analogous to our
treatment of groups with a nonabelian socle, we exploit the methods of [Lem23]
relying on algebraically independent invariants. The key result is the
following.

######  Theorem 6.1.

Let $G$ be a finite group, say of order $n$, and regard $G$ as a permutation
group of degree $n$ via its regular representation. Then there is an
algebraically independent set
$\\{f_{1},\dots,f_{n}\\}\subseteq\mathbb{Z}[x_{1},\dots,x_{n}]^{G}$ of
$G$-invariants satisfying:

  * •

$\deg f_{1}=1$;

  * •

$\deg f_{i}=2$ for $2\leq i\leq\frac{n+n_{2}+1}{2}$, where $n_{2}$ is the
number of elements of $G$ with order $2$; and

  * •

$\deg f_{i}=3$ for $\frac{n+n_{2}+1}{2}+1\leq i\leq n$.

This set is minimal, in the sense that any other set of algebraically
independent invariants $\\{f_{1}^{\prime},\dots,f_{n}^{\prime}\\}$ with $\deg
f_{i}^{\prime}\leq\deg f_{i+1}^{\prime}$ for $1\leq i\leq n-1$ must satisfy
$\deg f_{i}^{\prime}\geq\deg f_{i}$ for every $i\leq n$.

###### Proof.

We begin by describing the invariants $f_{i}$ explicitly. To this end, for a
polynomial $f\in\mathbb{Z}[x_{1},\dots,x_{n}]$, let $f^{G}:=\sum_{g\in
G}f^{g}$, which is necessarily $G$-invariant. We begin by setting
$f_{1}:=x_{1}^{G}=x_{1}+\dots+x_{n}$. We next consider invariants of the form
$(x_{1}x_{i})^{G}$ for $i\neq 1$, and claim that there are
$\frac{n+n_{2}-1}{2}$ distinct invariants of this form, which we take to be
the invariants $f_{2},\dots,f_{(n+n_{2}+1)/2}$. In particular, if
$(x_{1}x_{i})^{G}=(x_{1}x_{j})^{G}$ for some $i\neq j$, then there is some
$g\in G$ such that $(x_{1}x_{i})^{g}=x_{1}x_{j}$. Since $G$ acts regularly, we
cannot have $x_{1}^{g}=x_{1}$, so we must have that $x_{i}^{g}=x_{1}$ and
$x_{1}^{g}=x_{j}$. In particular, $g$ is the unique element of $G$ sending $i$
to $1$, and $j$ is the image of $1$ under the action of $g$. Write $g_{i}$ for
this element. Since $i\neq j$, we conclude that
$(x_{1}x_{i})^{G}=(x_{1}x_{j})^{G}$ for some $j\neq i$ precisely when $g_{i}$
has order greater than $2$. There are therefore
$\frac{n-1-n_{2}}{2}+n_{2}=\frac{n+n_{2}-1}{2}$ such invariants. Finally, for
those $i$ for which the element $g_{i}$ as above has order greater than $2$,
we also add the invariant $(x_{1}^{2}x_{i})^{G}$, and we take these invariants
to comprise the remaining $f_{i}$.

We next show that these invariants are algebraically independent, for which it
suffices to show that the determinant of the associated Jacobian matrix is a
non-zero polynomial. We therefore consider the partial derivatives of the
invariants above, and we claim that in the expression for the determinant as a
signed sum over permutations, there is a unique permutation giving rise to a
monomial with a maximal power of $x_{1}$. To see this, suppose first that $i$
is such that the element $g_{i}$ has order $2$, and let
$\phi=(x_{1}x_{i})^{G}$. In this case, the partial derivative
$\frac{\partial\phi}{\partial x_{i}}$ is the unique partial derivative
containing the monomial $x_{1}$. For $i\neq 1$ such that the element $g_{i}$
has order greater than $2$, let $\phi_{1}=(x_{i}x_{1})^{G}$ and
$\phi_{2}=(x_{1}^{2}x_{i})^{G}$. Unlike the case where $g_{i}$ has order $2$,
there are two partial derivatives of $\phi_{1}$ containing the monomial
$x_{1}$, namely $\frac{\partial\phi_{1}}{\partial x_{i}}$ and
$\frac{\partial\phi_{1}}{\partial x_{j}}$, where $j$ is the image of $1$ under
$g_{i}$. However, the partial derivative $\frac{\partial\phi_{2}}{\partial
x_{i}}$ is the only partial derivative of $\phi_{2}$ containing the monomial
$x_{1}^{2}$. It follows that, from the rows of the Jacobian corresponding to
$\phi_{1}$ and $\phi_{2}$, one must choose the partial derivatives
$\frac{\partial\phi_{1}}{\partial x_{j}}$ and
$\frac{\partial\phi_{2}}{\partial x_{i}}$ to obtain a maximal power of
$x_{1}$. Finally, since all other partial derivatives have been exhausted, it
follows that we must choose the partial derivative $\frac{\partial
f_{1}}{\partial x_{1}}$ of the invariant $f_{1}=(x_{1})^{G}$, yielding the
claim about the Jacobian and hence the theorem.

Finally, to see the claim about minimality, we observe that every monomial of