# Norm one tori and Hasse norm principle

Akinari Hoshi  Department of Mathematics, Niigata University, Niigata
950-2181, Japan<EMAIL_ADDRESS>,  Kazuki Kanai  Graduate
School of Science and Technology, Niigata University, Niigata 950-2181, Japan
<EMAIL_ADDRESS>and  Aiichi Yamasaki  Department of Mathematics,
Kyoto University, Kyoto 606-8502, Japan<EMAIL_ADDRESS>
###### Abstract.

Let $k$ be a field and $T$ be an algebraic $k$-torus. In 1969, over a global
field $k$, Voskresenskii proved that there exists an exact sequence $0\to
A(T)\to H^{1}(k,{\rm Pic}\,\overline{X})^{\vee}\to\Sha(T)\to 0$ where $A(T)$
is the kernel of the weak approximation of $T$, $\Sha(T)$ is the Shafarevich-
Tate group of $T$, $X$ is a smooth $k$-compactification of $T$,
$\overline{X}=X\times_{k}\overline{k}$, ${\rm Pic}\,\overline{X}$ is the
Picard group of $\overline{X}$ and $\vee$ stands for the Pontryagin dual. On
the other hand, in 1963, Ono proved that for the norm one torus
$T=R^{(1)}_{K/k}(\mathbbm{G}_{m})$ of $K/k$, $\Sha(T)=0$ if and only if the
Hasse norm principle holds for $K/k$. First, we determine $H^{1}(k,{\rm
Pic}\,\overline{X})$ for algebraic $k$-tori $T$ up to dimension $5$. Second,
we determine $H^{1}(k,{\rm Pic}\,\overline{X})$ for norm one tori
$T=R^{(1)}_{K/k}(\mathbbm{G}_{m})$ with $[K:k]=n\leq 15$ and $n\neq 12$. We
also show that $H^{1}(k,{\rm Pic}\,\overline{X})=0$ for
$T=R^{(1)}_{K/k}(\mathbbm{G}_{m})$ when the Galois group of the Galois closure
of $K/k$ is the Mathieu group $M_{n}\leq S_{n}$ with $n=11,12,22,23,24$.
Third, we give a necessary and sufficient condition for the Hasse norm
principle for $K/k$ with $[K:k]=n\leq 15$ and $n\neq 12$. As applications of
the results, we get the group $T(k)/R$ of $R$-equivalence classes over a local
field $k$ via Colliot-Thélène and Sansuc’s formula and the Tamagawa number
$\tau(T)$ over a number field $k$ via Ono’s formula
$\tau(T)=|H^{1}(k,\widehat{T})|/|\Sha(T)|$.

###### 2010 Mathematics Subject Classification:

Primary 11E72, 12F20, 13A50, 14E08, 20C10, 20G15.

Key words and phrases. Algebraic tori, norm one tori, Hasse norm principle,
weak approximation, rationality problem, flabby resolution.
This work was partially supported by JSPS KAKENHI Grant Numbers 16K05059,
19K03418, 20K03511. Parts of the work were finished when the first-named
author and the third-named author were visiting the National Center for
Theoretic Sciences (Taipei), whose support is gratefully acknowledged.

###### Contents

  1. 1 Introduction
  2. 2 Rationality problem for norm one tori
  3. 3 Strategy: flabby resolution of $G$-lattices
  4. 4 Proof of Theorem 1.5 and Theorem 1.6
  5. 5 Proof of Theorem 1.15 and Theorem 1.17
  6. 6 Proof of Theorem 1.18
  7. 7 Application $1$: $R$-equivalence in algebraic $k$-tori
  8. 8 Application $2$: Tamagawa number $\tau(T)$
  9. 9 Appendix: Computation of $H^{1}(G,J_{G/H})$ for $G={\rm Gal}(L/k)=nTm$ $(n\leq 15)$
  10. 10 GAP algorithms

##  1\. Introduction

Let $k$ be a field, $\overline{k}$ be a fixed separable closure of $k$ and
$\mathcal{G}={\rm Gal}(\overline{k}/k)$ be the absolute Galois group of $k$.
Let $T$ be an algebraic $k$-torus, i.e. a group $k$-scheme with fiber product
(base change) $T\times_{k}\overline{k}=T\times_{{\rm Spec}\,k}\,{\rm
Spec}\,\overline{k}\simeq(\mathbbm{G}_{m,\overline{k}})^{n}$; $k$-form of the
split torus $(\mathbbm{G}_{m})^{n}$. Then there exists the minimal (canonical)
finite Galois extension $K/k$ with Galois group $G={\rm Gal}(K/k)$ such that
$T$ splits over $K$: $T\times_{k}K\simeq(\mathbbm{G}_{m,K})^{n}$. It is also
well-known that there is the duality between the category of $G$-lattices,
i.e. finitely generated $\mathbbm{Z}[G]$-modules which are $\mathbbm{Z}$-free
as abelian groups, and the category of algebraic $k$-tori which split over $K$
(see Ono [Ono61, Section 1.2], Voskresenskii [Vos98, page 27, Example 6] and
Knus, Merkurjev, Rost and Tignol [KMRT98, page 333, Proposition 20.17]).
Indeed, if $T$ is an algebraic $k$-torus, then the character module
$\widehat{T}={\rm Hom}(T,\mathbbm{G}_{m})$ of $T$ may be regarded as a
$G$-lattice. Let $X$ be a smooth $k$-compactification of $T$, i.e. smooth
projective $k$-variety $X$ containing $T$ as a dense open subvariety, and
$\overline{X}=X\times_{k}\overline{k}$. There exists such a smooth
$k$-compactification of an algebraic $k$-torus $T$ over any field $k$ (due to
Hironaka [Hir64] for ${\rm char}\,k=0$, see Colliot-Thélène, Harari and
Skorobogatov [CTHS05, Corollaire 1] for any field $k$). A
$\mathcal{G}$-lattice $P$ is said to be permutation if $P$ has a
$\mathbbm{Z}$-basis permuted by $\mathcal{G}$ and a $\mathcal{G}$-lattice $F$
is said to be flabby (resp. coflabby) if $\widehat{H}^{-1}(\mathcal{H},F)=0$
(resp. $H^{1}(\mathcal{H},F)=0$) for any closed subgroup
$\mathcal{H}\leq\mathcal{G}$ where $\widehat{H}$ is the Tate cohomology.

######  Theorem 1.1 (Voskresenskii [Vos69, Section 4, page 1213], [Vos70,
Section 3, page 7], see also [Vos98, Section 4.6], [Kun07, Theorem 1.9],
[Vos74] and [CT07, Theorem 5.1, page 19] for any field $k$).

Let $k$ be a field and $\mathcal{G}={\rm Gal}(\overline{k}/k)$. Let $T$ be an
algebraic $k$-torus, $X$ be a smooth $k$-compactification of $T$ and
$\overline{X}=X\times_{k}\overline{k}$. Then there exists an exact sequence of
$\mathcal{G}$-lattices

$\displaystyle 0\to\widehat{T}\to\widehat{Q}\to{\rm Pic}\,\overline{X}\to 0$

where $\widehat{Q}$ is permutation and ${\rm Pic}\ \overline{X}$ is flabby.

We have $H^{1}(k,{\rm Pic}\,\overline{X})\simeq H^{1}(G,{\rm Pic}\,X_{K})$
where $K$ is the splitting field of $T$, $G={\rm Gal}(K/k)$ and
$X_{K}=X\times_{k}K$. Hence Theorem 1.1 says that for $G$-lattices
$M=\widehat{T}$ and $P=\widehat{Q}$, the exact sequence $0\to M\to P\to{\rm
Pic}\,X_{K}\to 0$ gives a flabby resolution of $M$ and the flabby class of $M$
is $[M]^{fl}=[{\rm Pic}\ X_{K}]$ as $G$-lattices (see Section 3, cf. Endo and
Miyata’s theorem [EM75, Lemma 1.1] (= Theorem 3.4 in the present paper)).

Let $k$ be a global field, i.e. a number field (a finite extension of
$\mathbbm{Q}$) or a function field of an algebraic curve over
$\mathbbm{F}_{q}$ (a finite extension of $\mathbbm{F}_{q}(t))$. Let $T$ be an
algebraic $k$-torus and $T(k)$ be the group of $k$-rational points of $T$.
Then $T(k)$ embeds into $\prod_{v\in V_{k}}T(k_{v})$ by the diagonal map where
$V_{k}$ is the set of all places of $k$ and $k_{v}$ is the completion of $k$
at $v$. Let $\overline{T(k)}$ be the closure of $T(k)$ in the product
$\prod_{v\in V_{k}}T(k_{v})$. The group

$\displaystyle A(T)=\left(\prod_{v\in V_{k}}T(k_{v})\right)/\overline{T(k)}$

is called the kernel of the weak approximation of $T$. We say that $T$ has the
weak approximation property if $A(T)=0$.

Let $E$ be a principal homogeneous space (= torsor) under $T$. Hasse principle
holds for $E$ means that if $E$ has a $k_{v}$-rational point for all $k_{v}$,
then $E$ has a $k$-rational point. The set $H^{1}(k,T)$ classifies all such
torsors $E$ up to (non-unique) isomorphism. We define the Shafarevich-Tate
group

$\displaystyle\Sha(T)={\rm Ker}\left\\{H^{1}(k,T)\xrightarrow{\rm
res}\bigoplus_{v\in V_{k}}H^{1}(k_{v},T)\right\\}.$

Then we say that $T$ satisfies the Hasse principle if $\Sha(T)=0$.

######  Theorem 1.2 (Voskresenskii [Vos69, Theorem 5, page 1213], [Vos70,
Theorem 6, page 9], see also [Vos98, Section 11.6, Theorem, page 120]).

Let $k$ be a global field, $T$ be an algebraic $k$-torus and $X$ be a smooth
$k$-compactification of $T$. Then there exists an exact sequence

$\displaystyle 0\to A(T)\to H^{1}(k,{\rm
Pic}\,\overline{X})^{\vee}\to\Sha(T)\to 0$

where $M^{\vee}={\rm Hom}(M,\mathbbm{Q}/\mathbbm{Z})$ is the Pontryagin dual
of $M$. Moreover, if $L$ is the splitting field of $T$ and $L/k$ is an
unramified extension, then $A(T)=0$ and $H^{1}(k,{\rm
Pic}\,\overline{X})^{\vee}\simeq\Sha(T)$.

For the last assertion, see [Vos98, Theorem, page 120]. It follows that
$H^{1}(k,{\rm Pic}\,\overline{X})=0$ if and only if $A(T)=0$ and $\Sha(T)=0$,
i.e. $T$ has the weak approximation property and satisfies the Hasse
principle. Theorem 1.2 was generalized to the case of linear algebraic groups
by Sansuc [San81].

The norm one torus $R^{(1)}_{K/k}(\mathbbm{G}_{m})$ of $K/k$ is the kernel of
the norm map $R_{K/k}(\mathbbm{G}_{m})\rightarrow\mathbbm{G}_{m}$ where
$R_{K/k}$ is the Weil restriction (see [Vos98, page 37, Section 3.12]). Such a
torus $R^{(1)}_{K/k}(\mathbbm{G}_{m})$ is biregularly isomorphic to the norm
hyper surface $f(x_{1},\ldots,x_{n})=1$ where $f\in k[x_{1},\ldots,x_{n}]$ is
the polynomial of total degree $n$ defined by the norm map
$N_{K/k}:K^{\times}\to k^{\times}$. When $K/k$ is Galois, we have that:

######  Theorem 1.3 (Voskresenskii [Vos70, Theorem 7], Colliot-Thélène and
Sansuc [CTS77, Proposition 1]).

Let $k$ be a field and $K/k$ be a finite Galois extension with Galois group
$G={\rm Gal}(K/k)$. Let $T=R^{(1)}_{K/k}(\mathbbm{G}_{m})$ be the norm one
torus of $K/k$ and $X$ be a smooth $k$-compactification of $T$. Then
$H^{1}(H,{\rm Pic}\,X_{K})\simeq H^{3}(H,\mathbbm{Z})$ for any subgroup $H$ of
$G$. In particular, $H^{1}(k,{\rm Pic}\,\overline{X})\simeq H^{1}(G,{\rm
Pic}\,X_{K})\simeq H^{3}(G,\mathbbm{Z})$ which is isomorphic to the Schur
multiplier $M(G)$ of $G$.

In other words, for $G$-lattice $J_{G}=\widehat{T}$,
$H^{1}(H,[J_{G}]^{fl})\simeq H^{3}(H,\mathbbm{Z})$ for any subgroup $H$ of $G$
and $H^{1}(G,[J_{G}]^{fl})\simeq H^{3}(G,\mathbbm{Z})\simeq
H^{2}(G,\mathbbm{Q}/\mathbbm{Z})$; the Schur multiplier of $G$. By the exact
sequence $0\to\mathbbm{Z}\to\mathbbm{Z}[G]\to J_{G}\to 0$, we also have
$\delta:H^{1}(G,J_{G})\simeq H^{2}(G,\mathbbm{Z})\simeq G^{ab}\simeq G/[G,G]$
where $\delta$ is the connecting homomorphism and $G^{ab}$ is the
abelianization of $G$ (for details, see Section 2).

Let $K$ be a finitely generated field extension of a field $k$. A field $K$ is
called rational over $k$ (or $k$-rational for short) if $K$ is purely
transcendental over $k$, i.e. $K$ is isomorphic to $k(x_{1},\ldots,x_{n})$,
the rational function field over $k$ with $n$ variables $x_{1},\ldots,x_{n}$
for some integer $n$. $K$ is called stably $k$-rational if
$K(y_{1},\ldots,y_{m})$ is $k$-rational for some algebraically independent
elements $y_{1},\ldots,y_{m}$ over $K$. Two fields $K$ and $K^{\prime}$ are
called stably $k$-isomorphic if $K(y_{1},\ldots,y_{m})\simeq
K^{\prime}(z_{1},\ldots,z_{n})$ over $k$ for some algebraically independent
elements $y_{1},\ldots,y_{m}$ over $K$ and $z_{1},\ldots,z_{n}$ over
$K^{\prime}$. When $k$ is an infinite field, $K$ is called retract
$k$-rational if there is a $k$-algebra $R$ contained in $K$ such that (i) $K$
is the quotient field of $R$, and (ii) the identity map $1_{R}:R\rightarrow R$
factors through a localized polynomial ring over $k$, i.e. there is an element
$f\in k[x_{1},\ldots,x_{n}]$, which is the polynomial ring over $k$, and there
are $k$-algebra homomorphisms $\varphi:R\rightarrow
k[x_{1},\ldots,x_{n}][1/f]$ and $\psi:k[x_{1},\ldots,x_{n}][1/f]\rightarrow R$
satisfying $\psi\circ\varphi=1_{R}$ (cf. [Sal84]). $K$ is called
$k$-unirational if $k\subset K\subset k(x_{1},\ldots,x_{n})$ for some integer
$n$. It is not difficult to see that “$k$-rational” $\Rightarrow$ “stably
$k$-rational” $\Rightarrow$ “retract $k$-rational” $\Rightarrow$
“$k$-unirational”.

An algebraic $k$-torus $T$ is said to be $k$-rational (resp. stably
$k$-rational, retract $k$-rational) if the function field $k(T)$ of $T$ is
$k$-rational (resp. stably $k$-rational, retract $k$-rational).

Note that an algebraic $k$-torus $T$ is always $k$-unirational (see [Vos98,
page 40, Example 21]). Tori of dimension $n$ over $k$ correspond bijectively
to the elements of the set $H^{1}(\mathcal{G},{\rm GL}_{n}(\mathbbm{Z}))$
where $\mathcal{G}={\rm Gal}(k_{\rm s}/k)$ since ${\rm
Aut}((\mathbbm{G}_{m})^{n})={\rm GL}_{n}(\mathbbm{Z})$. The algebraic
$k$-torus $T$ of dimension $n$ is determined uniquely by the integral
representation $h:\mathcal{G}\rightarrow{\rm GL}_{n}(\mathbbm{Z})$ up to
conjugacy, and the group $h(\mathcal{G})$ is a finite subgroup of ${\rm
GL}_{n}(\mathbbm{Z})$ (see [Vos98, page 57, Section 4.9])).

There are $2$ (resp. $13$, $73$, $710$, $6079$) $\mathbbm{Z}$-classes forming
$2$ (resp. $10$, $32$, $227$, $955$) $\mathbbm{Q}$-classes in ${\rm
GL}_{1}(\mathbbm{Z})$ (resp. ${\rm GL}_{2}(\mathbbm{Z})$, ${\rm
GL}_{3}(\mathbbm{Z})$, ${\rm GL}_{4}(\mathbbm{Z})$, ${\rm
GL}_{5}(\mathbbm{Z})$). It is easy to see that all the $1$-dimensional
algebraic $k$-tori $T$, i.e. the trivial torus $\mathbbm{G}_{m}$ and the norm
one torus $R^{(1)}_{K/k}(\mathbbm{G}_{m})$ of $K/k$ with $[K:k]=2$, are
$k$-rational. Voskresenskii [Vos67] proved that all the $13$ cases of
$2$-dimensional algebraic $k$-tori are $k$-rational. Note that $T$ is retract
$k$-rational $\Rightarrow$ $H^{1}(k,{\rm Pic}\,\overline{X})=0$ and over
global field $k$, $H^{1}(k,{\rm Pic}\,\overline{X})=0\Rightarrow
A(T)=\Sha(T)=0$ (see Section 3 and also Manin [Man74, §30]).

Kunyavskii [Kun90] solved the rationality problem for $3$-dimensional
algebraic $k$-tori. In the classification, there exist $73$ cases of
$3$-dimensional algebraic $k$-tori which correspond to $73$
$\mathbbm{Z}$-conjugacy classes of finite subgroups of ${\rm
GL}_{3}(\mathbbm{Z})$, and $15$ cases of them are not $k$-rational (resp. not
stably $k$-rational, not retract $k$-rational). Using the classification,
Kunyavskii [Kun84] showed that only $2$ cases of algebraic $k$-tori of
dimension $3$ satisfy the non-vanishing $H^{1}(k,{\rm Pic}\,\overline{X})\neq
0$ among $15$ cases of non-rational $k$-tori. These two $k$-tori are norm one
tori $T=R^{(1)}_{K/k}(\mathbbm{G}_{m})$ with $[K:k]=4$:

######  Theorem 1.4 (Kunyavskii [Kun84, Proposition 1]).

Let $k$ be a field, $T$ be an algebraic $k$-torus of dimension $3$ and $X$ be
a smooth $k$-compactification of $T$. Then, among $15$ cases of non-rational
algebraic $k$-tori $T$,

$\displaystyle H^{1}(k,{\rm
Pic}\,\overline{X})=\begin{cases}\mathbbm{Z}/2\mathbbm{Z}&{\rm if}\
T=R^{(1)}_{K_{1}/k}(\mathbbm{G}_{m})\ {\rm or}\
R^{(1)}_{K_{2}/k}(\mathbbm{G}_{m})\\\ 0&{\rm otherwise}\end{cases}$

where $K_{1}/k$ $($resp. $K_{2}/k$$)$ is a field extension of degree $4$ whose
Galois closure $L_{1}/k$ $($resp. $L_{2}/k$$)$ satisfies ${\rm
Gal}(L_{1}/k)\simeq V_{4}$; the Klein four group $($resp. ${\rm
Gal}(L_{2}/k)\simeq A_{4}$; the alternating group of degree $4$$)$. In
particular, if $k$ is a global field, then $A(T)=\Sha(T)=0$ except for
$T=R^{(1)}_{K_{1}/k}(\mathbbm{G}_{m})$ and
$T=R^{(1)}_{K_{2}/k}(\mathbbm{G}_{m})$.

Hoshi and Yamasaki [HY17] classified stably/retract $k$-rational algebraic
$k$-tori of dimensions $4$ and $5$. In the classification, there exist $710$
(resp. $6079$) cases of $4$-dimensional (resp. $5$-dimensional) algebraic
$k$-tori which correspond to $710$ (resp. $6079$) $\mathbbm{Z}$-conjugacy
classes of finite subgroups of ${\rm GL}_{4}(\mathbbm{Z})$ (resp. ${\rm
GL}_{5}(\mathbbm{Z})$), and $216$ (resp. $3003$) cases of them are not retract
$k$-rational.

The first main result (Theorem 1.5 and Theorem 1.6) of this paper is to
classify the algebraic $k$-tori $T$ with non-vanishing $H^{1}(k,{\rm
Pic}\,\overline{X})\neq 0$ in dimensions $4$ and $5$:

######  Theorem 1.5 (see Theorem 4.1 for the detailed statement).

Let $k$ be a field, $T$ be an algebraic $k$-torus of dimension $4$ and $X$ be
a smooth $k$-compactification of $T$. Among $216$ cases of not retract
rational algebraic $k$-tori $T$, there exist exactly $2$ $($resp. $20$,
$194$$)$ algebraic $k$-tori with $H^{1}(k,{\rm
Pic}\,\overline{X})\simeq(\mathbbm{Z}/2\mathbbm{Z})^{\oplus 2}$ $($resp.
$H^{1}(k,{\rm Pic}\,\overline{X})\simeq\mathbbm{Z}/2\mathbbm{Z}$,
$H^{1}(k,{\rm Pic}\,\overline{X})=0$$)$.

######  Theorem 1.6 (see Theorem 4.2 for the detailed statement).

Let $k$ be a field, $T$ be an algebraic $k$-torus of dimension $5$ and $X$ be
a smooth $k$-compactification of $T$. Among $3003$ cases of not retract
rational algebraic $k$-tori $T$, there exist exactly $11$ $($resp. $263$,
$2729$$)$ algebraic $k$-tori with $H^{1}(k,{\rm
Pic}\,\overline{X})\simeq(\mathbbm{Z}/2\mathbbm{Z})^{\oplus 2}$ $($resp.
$H^{1}(k,{\rm Pic}\,\overline{X})\simeq\mathbbm{Z}/2\mathbbm{Z}$,
$H^{1}(k,{\rm Pic}\,\overline{X})=0$$)$.

Note that Hoshi and Yamasaki [HY17, Chapter 7] showed the vanishing
$H^{1}(k,{\rm Pic}\,\overline{X})\simeq H^{1}(G,[\widehat{T}]^{fl})=0$ for any
Bravais group $G$ of dimension $n\leq 6$ (see also [Vos83], [Vos98, Section
8]). There exists $1$ (resp. $5$, $14$, $64$, $189$, $841$) Bravais group of
dimension $n=1$ (resp. $2$, $3$, $4$, $5$, $6$) (see [HY17, Example 4.16]).

Let $G$ be a finite group and $M$ be a $G$-lattice. We define

$\displaystyle{}^{i}_{\omega}(G,M):={\rm
Ker}\left\\{H^{i}(G,M)\xrightarrow{{\rm res}}\bigoplus_{g\in G}H^{i}(\langle
g\rangle,M)\right\\}\quad(i\geq 1).$

The following is a theorem of Colliot-Thélène and Sansuc [CTS87]:

######  Theorem 1.7 (Colliot-Thélène and Sansuc [CTS87, Proposition 9.5 (ii)],
see also [San81, Proposition 9.8] and [Vos98, page 98]).

Let $k$ be a field with ${\rm char}\,k=0$ and $K/k$ be a finite Galois
extension with Galois group $G={\rm Gal}(K/k)$. Let $T$ be an algebraic
$k$-torus which splits over $K$ and $X$ be a smooth $k$-compactification of
$T$. Then we have

$\displaystyle{}^{2}_{\omega}(G,\widehat{T})\simeq H^{1}(G,{\rm
Pic}\,X_{K})\simeq{\rm Br}(X)/{\rm Br}(k)$

where ${\rm Br}(X)$ is the étale cohomological Brauer Group of $X$ $($it is
the same as the Azumaya-Brauer group of $X$ for such $X$, see [CTS87, page
199]$)$.

In other words, for $G$-lattice $M=\widehat{T}$, we have $H^{1}(k,{\rm
Pic}\,\overline{X})\simeq H^{1}(G,{\rm Pic}\,X_{K})\simeq
H^{1}(G,[M]^{fl})\simeq{}^{2}_{\omega}(G,M)\simeq{\rm Br}(X)/{\rm Br}(k)$ (for
the flabby class $[M]^{fl}$ of $M$, see Section 3). Hence Theorem 1.4, Theorem
1.5 and Theorem 1.6 compute
$H^{1}(G,[M]^{fl})\simeq{}^{2}_{\omega}(G,M)\simeq{\rm Br}(X)/{\rm Br}(k)$
where $M=\widehat{T}$. We also see ${\rm Br}_{\rm nr}(k(X)/k)={\rm
Br}(X)\subset{\rm Br}(k(X))$ (see Colliot-Thélène [CTS07, Theorem 5.11],
Saltman [Sal99, Proposition 10.5]).

Let $k$ be a global field, $K/k$ be a finite extension and
$\mathbbm{A}_{K}^{\times}$ be the idele group of $K$. We say that the Hasse
norm principle holds for $K/k$ if $(N_{K/k}(\mathbbm{A}_{K}^{\times})\cap
k^{\times})/N_{K/k}(K^{\times})=1$ where $N_{K/k}$ is the norm map.

Hasse [Has31, Satz, page 64] proved that the Hasse norm principle holds for
any cyclic extension $K/k$ but does not hold for bicyclic extension
$\mathbbm{Q}(\sqrt{-39},\sqrt{-3})/\mathbbm{Q}$. For Galois extensions $K/k$,
Tate [Tat67] gave the following theorem:

######  Theorem 1.8 (Tate [Tat67, page 198]).

Let $k$ be a global field, $K/k$ be a finite Galois extension with Galois
group ${\rm Gal}(K/k)\simeq G$. Let $V_{k}$ be the set of all places of $k$
and $G_{v}$ be the decomposition group of $G$ at $v\in V_{k}$. Then we have

$\displaystyle(N_{K/k}(\mathbbm{A}_{K}^{\times})\cap
k^{\times})/N_{K/k}(K^{\times})\simeq{\rm Coker}\left\\{\bigoplus_{v\in
V_{k}}\widehat{H}^{-3}(G_{v},\mathbbm{Z})\xrightarrow{\rm
cores}\widehat{H}^{-3}(G,\mathbbm{Z})\right\\}$

where $\widehat{H}$ is the Tate cohomology. In particular, the Hasse norm
principle holds for $K/k$ if and only if the restriction map
$H^{3}(G,\mathbbm{Z})\xrightarrow{\rm res}\bigoplus_{v\in
V_{k}}H^{3}(G_{v},\mathbbm{Z})$ is injective.

Let $S_{n}$ (resp. $A_{n}$, $D_{n}$, $C_{n}$) be the symmetric (resp. the
alternating, the dihedral, the cyclic) group of degree $n$ of order $n!$
(resp. $n!/2$, $2n$, $n$). Let $V_{4}\simeq C_{2}\times C_{2}$ be the Klein
four group.

If $G\simeq C_{n}$ is cyclic, then $\widehat{H}^{-3}(G,\mathbbm{Z})\simeq
H^{3}(G,\mathbbm{Z})\simeq H^{1}(G,\mathbbm{Z})=0$ and hence the Hasse’s
original theorem follows. If there exists a place $v$ of $k$ such that
$G_{v}=G$, then the Hasse norm principle also holds for $K/k$. For example,
the Hasse norm principle holds for $K/k$ with $G\simeq V_{4}$ if and only if
there exists a place $v$ of $k$ such that $G_{v}=V_{4}$ because
$H^{3}(V_{4},\mathbbm{Z})\simeq\mathbbm{Z}/2\mathbbm{Z}$ and
$H^{3}(C_{2},\mathbbm{Z})=0$. The Hasse norm principle holds for $K/k$ with
$G\simeq(C_{2})^{3}$ if and only if (i) there exists a place $v$ of $k$ such
that $G_{v}=G$ or (ii) there exist places $v_{1},v_{2},v_{3}$ of $k$ such that
$G_{v_{i}}\simeq V_{4}$ and
$H^{3}(G,\mathbbm{Z})=H^{3}(G_{v_{1}},\mathbbm{Z})\oplus
H^{3}(G_{v_{2}},\mathbbm{Z})\oplus H^{3}(G_{v_{3}},\mathbbm{Z})$ as inner
direct sum because
$H^{3}(G,\mathbbm{Z})\simeq(\mathbbm{Z}/2\mathbbm{Z})^{\oplus 3}$ and
$H^{3}(V_{4},\mathbbm{Z})\simeq\mathbbm{Z}/2\mathbbm{Z}$.

Ono [Ono63] established the relationship between the Hasse norm principle for
$K/k$ and the Hasse principle for the norm one torus
$R^{(1)}_{K/k}(\mathbbm{G}_{m})$ of $K/k$:

######  Theorem 1.9 (Ono [Ono63, page 70], see also Platonov [Pla82, page 44],
Kunyavskii [Kun84, Remark 3], Platonov and Rapinchuk [PR94, page 307]).

Let $k$ be a global field and $K/k$ be a finite extension. Then

$\displaystyle\Sha(R^{(1)}_{K/k}(\mathbbm{G}_{m}))\simeq(N_{K/k}(\mathbbm{A}_{K}^{\times})\cap
k^{\times})/N_{K/k}(K^{\times}).$

In particular, $\Sha(R^{(1)}_{K/k}(\mathbbm{G}_{m}))=0$ if and only if the
Hasse norm principle holds for $K/k$.

The Hasse norm principle for Galois extensions $K/k$ was investigated by Gerth
[Ger77], [Ger78] and Gurak [Gur78a], [Gur78b], [Gur80] (see also [PR94, pages
308–309]), etc. Gurak [Gur78a] showed that the Hasse norm principle holds for
Galois extension $K/k$ if all the Sylow subgroups of ${\rm Gal}(K/k)$ are
cyclic. Note that this also follows from Theorem 1.9 and the retract
$k$-rationality of $T=R^{(1)}_{K/k}(\mathbbm{G}_{m})$ due to Endo and Miyata
[EM75, Theorem 2.3].

However, for non-Galois extension $K/k$, very little is known about the Hasse
norm principle. Bartels [Bar81a] (resp. [Bar81b]) showed that the Hasse norm
principle for $K/k$ holds when $[K:k]$ is prime (resp. ${\rm Gal}(L/k)\simeq
D_{n}$). The former case also follows from Theorem 1.9 and the retract
$k$-rationality of $T=R^{(1)}_{K/k}(\mathbbm{G}_{m})$ due to Colliot-Thélène
and Sansuc [CTS87, Proposition 9.1] (see Theorem 2.1).

######  Theorem 1.10 (Voskresenskii and Kunyavskii [VK84], see also
Voskresenskii [Vos88, Theorem 4, Corollary]).

Let $k$ be a number field, $K/k$ be a finite extension of degree $n$ and $L/k$
be the Galois closure of $K/k$ with ${\rm Gal}(L/k)\simeq S_{n}$; the
symmetric group of degree $n$. Let $T=R^{(1)}_{K/k}(\mathbbm{G}_{m})$ be the
norm one torus of $K/k$ and $X$ be a smooth $k$-compactification of $T$. Then
$H^{1}(S_{n},{\rm Pic}\,X_{L})=0$. In particular, $T$ has the weak
approximation property and the Hasse norm principle holds for $K/k$.

######  Theorem 1.11 (Macedo [Mac20]).

Let $k$ be a number field, $K/k$ be a finite extension of degree $n\geq 5$ and
$L/k$ be the Galois closure of $K/k$ with ${\rm Gal}(L/k)\simeq A_{n}$; the
alternating group of degree $n\geq 5$. Let $T=R^{(1)}_{K/k}(\mathbbm{G}_{m})$
be the norm one torus of $K/k$. Then ${}^{2}_{\omega}(A_{n},\widehat{T})=0$.
In particular, $T$ has the weak approximation property and the Hasse norm
principle holds for $K/k$.

######  Remark 1.12.

Applying Theorem 1.2 to $T=R^{(1)}_{K/k}(\mathbbm{G}_{m})$, it follows from
Theorem 1.9 that $H^{1}(k,{\rm Pic}\,\overline{X})=0$ if and only if $A(T)=0$
and $\Sha(T)=0$, i.e. $T$ has the weak approximation property and the Hasse
norm principle holds for $K/k$. In the algebraic language, the latter
condition $\Sha(T)=0$ means that for the corresponding norm hyper surface
$f(x_{1},\ldots,x_{n})=b$, it has a $k$-rational point if and only if it has a
$k_{v}$-rational point for any valuation $v$ of $k$ where $f\in
k[x_{1},\ldots,x_{n}]$ is the polynomial of total degree $n$ defined by the
norm map $N_{K/k}:K^{\times}\to k^{\times}$ and $b\in k^{\times}$ (see [Vos98,
Example 4, page 122]).

Let $nTm$ be the $m$-th transitive subgroup of $S_{n}$ up to conjugacy (see
Butler and McKay [BM83], [GAP]).

Let $k$ be a number field, $K/k$ be a field extension of degree $n$ and $L/k$
be the Galois closure of $K/k$ with ${\rm Gal}(L/k)\simeq G$. Then we may
regard $G$ as the transitive subgroup $G=nTm\leq S_{n}$. Let $v$ be a place of
$k$ and $G_{v}$ be the decomposition group of $G$ at $v$. Using Theorem 1.4,
Kunyavskii [Kun84] gave a necessary and sufficient condition for the Hasse
norm principle for $n=4$:

######  Theorem 1.13 (Kunyavskii [Kun84, page 1899]).

Let $k$ be a number field, $K/k$ be a field extension of degree $4$ and $L/k$
be the Galois closure of $K/k$. Let $G={\rm Gal}(L/k)=4Tm$ $(1\leq m\leq 5)$
be a transitive subgroup of $S_{4}$ and $H={\rm Gal}(L/K)$ with $[G:H]=4$. Let
$T=R^{(1)}_{K/k}(\mathbbm{G}_{m})$ be the norm one torus of $K/k$. Then
$A(T)=\Sha(T)=0$ except for $4T2\simeq V_{4}$ and $4T4\simeq A_{4}$. For
$4T2\simeq V_{4}$ and $4T4\simeq A_{4}$, either (i) $A(T)=0$ and
$\Sha(T)\simeq\mathbbm{Z}/2\mathbbm{Z}$ or (ii)
$A(T)=\mathbbm{Z}/2\mathbbm{Z}$ and $\Sha(T)=0$, and the following conditions
are equivalent:
(ii) $A(T)=\mathbbm{Z}/2\mathbbm{Z}$ and $\Sha(T)=0$;
(iii) there exists a place $v$ of $k$ $($which ramifies in $L$$)$ such that
$V_{4}\leq G_{v}$.

Drakokhrust and Platonov [DP87] gave a necessary and sufficient condition for
the Hasse norm principle for $n=6$ ($G=6Tm\ (1\leq m\leq 16))$:

######  Theorem 1.14 (Drakokhrust and Platonov [DP87, Lemma 12, Proposition 6,
Lemma 13]).

Let $k$ be a number field, $K/k$ be a field extension of degree $6$ and $L/k$
be the Galois closure of $K/k$. Let $G={\rm Gal}(L/k)=6Tm$ $(1\leq m\leq 16)$
be a transitive subgroup of $S_{6}$ and $H={\rm Gal}(L/K)$ with $[G:H]=6$. Let
$T=R^{(1)}_{K/k}(\mathbbm{G}_{m})$ be the norm one torus of $K/k$. Then
$\Sha(T)=0$ except for $6T4\simeq A_{4}$ and $6T12\simeq A_{5}$. For
$6T4\simeq A_{4}$ and $6T12\simeq A_{5}$, (i)
$\Sha(T)\leq\mathbbm{Z}/2\mathbbm{Z}$; and (ii) $\Sha(T)=0$ if and only if
there exists a place $v$ of $k$ $($which ramifies in $L$$)$ such that
$V_{4}\leq G_{v}$.

The number of transitive subgroups $nTm$ of $S_{n}$ $(2\leq n\leq 15)$ up to
conjugacy is given as follows (see Butler and McKay [BM83] for $n\leq 11$,
Royle [Roy87] for $n=12$, Butler [But93] for $n=14,15$ and [GAP]):

$n$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ | $13$ | $14$ | $15$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\\#$ of $nTm$ | $1$ | $2$ | $5$ | $5$ | $16$ | $7$ | $50$ | $34$ | $45$ | $8$ | $301$ | $9$ | $63$ | $104$

The following theorem which is one of the main results of this paper
classifies the norm one tori $T=R^{(1)}_{K/k}(\mathbbm{G}_{m})$ with non-
vanishing $H^{1}(k,{\rm Pic}\,\overline{X})\neq 0$ for $[K:k]=n\leq 15$ and
$n\neq 12$.

######  Theorem 1.15.

Let $2\leq n\leq 15$ be an integer with $n\neq 12$. Let $k$ be a field, $K/k$
be a separable field extension of degree $n$ and $L/k$ be the Galois closure
of $K/k$. Assume that $G={\rm Gal}(L/k)=nTm$ is a transitive subgroup of
$S_{n}$ and $H={\rm Gal}(L/K)$ with $[G:H]=n$. Let
$T=R^{(1)}_{K/k}(\mathbbm{G}_{m})$ be the norm one torus of $K/k$ of dimension
$n-1$ and $X$ be a smooth $k$-compactification of $T$. Then $H^{1}(k,{\rm
Pic}\,\overline{X})\neq 0$ if and only if $G$ is given as in Table $1$. In
particular, if $k$ is a number field and $L/k$ is an unramified extension,
then $A(T)=0$ and $H^{1}(k,{\rm Pic}\,\overline{X})\simeq\Sha(T)$.

Table $1$: $H^{1}(k,{\rm Pic}\,\overline{X})\simeq H^{1}(G,[J_{G/H}]^{fl})\neq
0$ where $G=nTm$ with $2\leq n\leq 15$ and $n\neq 12$

$G$ |  $H^{1}(k,{\rm Pic}\,\overline{X})$ $\simeq H^{1}(G,[J_{G/H}]^{fl})$
---|---
$4T2\simeq V_{4}$ | $\mathbbm{Z}/2\mathbbm{Z}$
$4T4\simeq A_{4}$ | $\mathbbm{Z}/2\mathbbm{Z}$
$6T4\simeq A_{4}$ | $\mathbbm{Z}/2\mathbbm{Z}$
$6T12\simeq A_{5}$ | $\mathbbm{Z}/2\mathbbm{Z}$
$8T2\simeq C_{4}\times C_{2}$ | $\mathbbm{Z}/2\mathbbm{Z}$
$8T3\simeq(C_{2})^{3}$ | $(\mathbbm{Z}/2\mathbbm{Z})^{\oplus 3}$
$8T4\simeq D_{4}$ | $\mathbbm{Z}/2\mathbbm{Z}$
$8T9\simeq D_{4}\times C_{2}$ | $\mathbbm{Z}/2\mathbbm{Z}$
$8T11\simeq(C_{4}\times C_{2})\rtimes C_{2}$ | $\mathbbm{Z}/2\mathbbm{Z}$
$8T13\simeq A_{4}\times C_{2}$ | $\mathbbm{Z}/2\mathbbm{Z}$
$8T14\simeq S_{4}$ | $\mathbbm{Z}/2\mathbbm{Z}$
$8T15\simeq C_{8}\rtimes V_{4}$ | $\mathbbm{Z}/2\mathbbm{Z}$
$8T19\simeq(C_{2})^{3}\rtimes C_{4}$ | $\mathbbm{Z}/2\mathbbm{Z}$
$8T21\simeq(C_{2})^{3}\rtimes C_{4}$ | $\mathbbm{Z}/2\mathbbm{Z}$
$8T22\simeq(C_{2})^{3}\rtimes V_{4}$ | $\mathbbm{Z}/2\mathbbm{Z}$
$8T31\simeq((C_{2})^{4}\rtimes C_{2})\rtimes C_{2}$ | $\mathbbm{Z}/2\mathbbm{Z}$
$8T32\simeq((C_{2})^{3}\rtimes V_{4})\rtimes C_{3}$ | $\mathbbm{Z}/2\mathbbm{Z}$
$8T37\simeq{\rm PSL}_{3}(\mathbbm{F}_{2})\simeq{\rm PSL}_{2}(\mathbbm{F}_{7})$ | $\mathbbm{Z}/2\mathbbm{Z}$
$8T38\simeq(((C_{2})^{4}\rtimes C_{2})\rtimes C_{2})\rtimes C_{3}$ | $\mathbbm{Z}/2\mathbbm{Z}$
$9T2\simeq(C_{3})^{2}$ | $\mathbbm{Z}/3\mathbbm{Z}$
$9T5\simeq(C_{3})^{2}\rtimes C_{2}$ | $\mathbbm{Z}/3\mathbbm{Z}$
$9T7\simeq(C_{3})^{2}\rtimes C_{3}$ | $\mathbbm{Z}/3\mathbbm{Z}$
$9T9\simeq(C_{3})^{2}\rtimes C_{4}$ | $\mathbbm{Z}/3\mathbbm{Z}$
$9T11\simeq(C_{3})^{2}\rtimes C_{6}$ | $\mathbbm{Z}/3\mathbbm{Z}$
$9T14\simeq(C_{3})^{2}\rtimes Q_{8}$ | $\mathbbm{Z}/3\mathbbm{Z}$
$9T23\simeq((C_{3})^{2}\rtimes Q_{8})\rtimes C_{3}$ | $\mathbbm{Z}/3\mathbbm{Z}$
$10T7\simeq A_{5}$ | $\mathbbm{Z}/2\mathbbm{Z}$
$10T26\simeq{\rm PSL}_{2}(\mathbbm{F}_{9})\simeq A_{6}$ | $\mathbbm{Z}/2\mathbbm{Z}$
$10T32\simeq S_{6}$ | $\mathbbm{Z}/2\mathbbm{Z}$
$14T30\simeq{\rm PSL}_{2}(\mathbbm{F}_{13})$ | $\mathbbm{Z}/2\mathbbm{Z}$
$15T9\simeq(C_{5})^{2}\rtimes C_{3}$ | $\mathbbm{Z}/5\mathbbm{Z}$
$15T14\simeq(C_{5})^{2}\rtimes S_{3}$ | $\mathbbm{Z}/5\mathbbm{Z}$

######  Remark 1.16.

In Table $1$, only the abelian groups of prime exponent $p$ appear as
$H^{1}(k,{\rm Pic}\,\overline{X})$. However, we find that $H^{1}(k,{\rm
Pic}\,\overline{X})\simeq\mathbbm{Z}/4\mathbbm{Z}$ for
$G=12T31\simeq(C_{4})^{2}\rtimes C_{3}$ and $G=12T57\simeq((C_{4}\times
C_{2})\rtimes C_{4})\rtimes C_{3}$ by using the same technique as in the proof
of Theorem 1.15.

Additionally, by using the same method of Theorem 1.15, we obtain the
vanishing $H^{1}(k,{\rm Pic}\,\overline{X})=0$ for the $5$ Mathieu groups
$M_{n}\leq S_{n}$ where $n=11,12,22,23,24$ (see Dixon and Mortimer [DM96,
Chapter 6], Gorenstein, Lyons and Solomon [GLS98, Chapter 5] for the $5$
Mathieu groups):

######  Theorem 1.17.

Let $k$ be a field, $K/k$ be a separable field extension of degree $n$ and
$L/k$ be the Galois closure of $K/k$. Assume that $G={\rm Gal}(L/k)=M_{n}\leq
S_{n}$ $(n=11,12,22,23,24)$ is the Mathieu group of degree $n$ and $H={\rm
Gal}(L/K)$ with $[G:H]=n$. Let $T=R^{(1)}_{K/k}(\mathbbm{G}_{m})$ be the norm
one torus of $K/k$ of dimension $n-1$ and $X$ be a smooth $k$-compactification
of $T$. Then $H^{1}(k,{\rm Pic}\,\overline{X})=0$. In particular, if $k$ is a
number field, then $A(T)=0$ and $\Sha(T)=0$.

Let $Z(G)$ be the center of a group $G$, $[G,G]$ be the commutator group of
$G$ and ${\rm Syl}_{p}(G)$ be a $p$-Sylow subgroup of $G$ where $p$ is a
prime. Let ${\rm Orb}_{G}(i)$ be the orbit of $1\leq i\leq n$ under the action
of $G\leq S_{n}$.

By Theorem 1.15, we obtain the following theorem which gives a necessary and
sufficient condition for the Hasse norm principle for $K/k$ where $[K:k]=n\leq
15$ and $n\neq 12$. Note that a place $v$ of $k$ with non-cyclic decomposition
group $G_{v}$ as in Theorem 1.18 must be ramified in $L$ because if $v$ is
unramified, then $G_{v}$ is cyclic.

######  Theorem 1.18.

Let $2\leq n\leq 15$ be an integer with $n\neq 12$. Let $k$ be a number field,
$K/k$ be a field extension of degree $n$ and $L/k$ be the Galois closure of
$K/k$. Assume that $G={\rm Gal}(L/k)=nTm$ is a transitive subgroup of $S_{n}$,
$H={\rm Gal}(L/K)$ with $[G:H]=n$ and $G_{v}$ is the decomposition group of
$G$ at a place $v$ of $k$. Let $T=R^{(1)}_{K/k}(\mathbbm{G}_{m})$ be the norm
one torus of $K/k$ of dimension $n-1$ and $X$ be a smooth $k$-compactification
of $T$. Then $A(T)=\Sha(T)=0$ except for the cases in Table $1$. For the cases
in Table $1$ except for $G=8T3$, either (a) $A(T)=0$ and $\Sha(T)\simeq
H^{1}(k,{\rm Pic}\,\overline{X})$ or (b) $A(T)\simeq H^{1}(k,{\rm
Pic}\,\overline{X})$ and $\Sha(T)=0$. For $G=8Tm$ $(m=9,11,15,19,22,32)$, we
assume that $H$ is the stabilizer of one of the letters in $G$. Then a
necessary and sufficient condition for $\Sha(T)=0$ is given as in Table $2$.

Table $2$: $\Sha(T)=0$ for $T=R^{(1)}_{K/k}(\mathbbm{G}_{m})$ and $G={\rm
Gal}(L/k)=nTm$ as in Table $1$

$G$ |  $\Sha(T)=0$ if and only if there exists a place $v$ of $k$ such that
---|---
$4T2\simeq V_{4}$ | $V_{4}\leq G_{v}$ (Tate [Tat67] for $4T2\simeq V_{4}$) (Kunyavskii [Kun84] for $4T4\simeq A_{4}$)
$4T4\simeq A_{4}$
$6T4\simeq A_{4}$ | $V_{4}\leq G_{v}$ (Drakokhrust and Platonov [DP87])
$6T12\simeq A_{5}$
$8T3\simeq(C_{2})^{3}$ | (Tate [Tat67], see the second paragraph after Theorem 1.8)
$8T4\simeq D_{4}$ | $V_{4}\leq G_{v}$ (Tate [Tat67] for $8T4\simeq D_{4}$)
$8T13\simeq A_{4}\times C_{2}$
$8T14\simeq S_{4}$
$8T37\simeq{\rm PSL}_{2}(\mathbbm{F}_{7})$
$8T2\simeq C_{4}\times C_{2}$ | $G_{v}=G$ (Tate [Tat67] for $8T2\simeq C_{4}\times C_{2}$)
$8T21\simeq(C_{2})^{3}\rtimes C_{4}$
|  (i) $V_{4}\leq G_{v}$ where $V_{4}\cap[{\rm Syl}_{2}(G),{\rm
Syl}_{2}(G)]=1$ with ${\rm Syl}_{2}(G)\lhd G$ $($equivalently, $|{\rm
Orb}_{V_{4}}(i)|=4$ for any $1\leq i\leq 8$ and $V_{4}\cap Z(G)=1$$)$, (ii)
$C_{4}\times C_{2}\leq G_{v}$ where $(C_{4}\times C_{2})\cap[{\rm
Syl}_{2}(G),{\rm Syl}_{2}(G)]\simeq C_{2}$ $($equivalently, $C_{4}\times
C_{2}$ is transitive in $S_{8}$$)$ or (iii) $(C_{2})^{3}\rtimes C_{4}\leq
G_{v}$
$8T31\simeq(C_{2})^{4}\rtimes V_{4}$
$8T38\simeq 8T31\rtimes C_{3}$
$8T9\simeq D_{4}\times C_{2}$ |  (i) $V_{4}\leq G_{v}$ where $|{\rm Orb}_{V_{4}}(i)|=4$ for any $1\leq i\leq 8$ and $V_{4}\cap[G,G]=1$; or (ii) $C_{4}\times C_{2}\leq G_{v}$
$8T11\simeq Q_{8}\rtimes C_{2}$ |  $C_{4}\times C_{2}\leq G_{v}$ where $C_{4}\times C_{2}$ is transitive in $S_{8}$
$8T15\simeq C_{8}\rtimes V_{4}$ |  (i) $V_{4}\leq G_{v}$ where $|{\rm Orb}_{V_{4}}(i)|=2$ for any $1\leq i\leq 8$ and $V_{4}\cap[G,G]=1$ $($equivalently, $|{\rm Orb}_{V_{4}}(i)|=2$ for any $1\leq i\leq 8$ and $V_{4}$ is not in $A_{8}$$)$ or (ii) $C_{4}\times C_{2}\leq G_{v}$ where $(C_{4}\times C_{2})\cap[G,G]\simeq C_{2}$ $($equivalently, $C_{4}\times C_{2}$ is transitive in $S_{8}$$)$
$8T19\simeq(C_{2})^{3}\rtimes C_{4}$ |  (i) $V_{4}\leq G_{v}$ where $V_{4}\cap Z(G)=1$ and $V_{4}\cap Z^{2}(G)\simeq C_{2}$ with the upper central series $1\leq Z(G)\leq Z^{2}(G)\leq G$ $($equivalently, $|{\rm Orb}_{V_{4}}(i)|=4$ for any $1\leq i\leq 8$ and $V_{4}\cap Z(G)=1$$)$; or (ii) $C_{4}\times C_{2}\leq G_{v}$ where $C_{4}\times C_{2}$ is not transitive in $S_{8}$ or $[G,G]\leq C_{4}\times C_{2}$
$8T22\simeq(C_{2})^{3}\rtimes V_{4}$ |  (i) $V_{4}\leq G_{v}$ where $|{\rm Orb}_{V_{4}}(i)|=4$ for any $1\leq i\leq 8$ and $V_{4}\cap Z(G)=1$ or (ii) $C_{4}\times C_{2}\leq G_{v}$ where $C_{4}\times C_{2}$ is transitive in $S_{8}$
$8T32\simeq 8T22\rtimes C_{3}$
$9T2\simeq(C_{3})^{2}$ |
$9T5\simeq(C_{3})^{2}\rtimes C_{2}$ |
$9T7\simeq(C_{3})^{2}\rtimes C_{3}$ |
$9T9\simeq(C_{3})^{2}\rtimes C_{4}$ |  $(C_{3})^{2}\leq G_{v}$ (Tate [Tat67] for $9T2\simeq(C_{3})^{2}$)
$9T11\simeq(C_{3})^{2}\rtimes C_{6}$ |
$9T14\simeq(C_{3})^{2}\rtimes Q_{8}$ |
$9T23\simeq 9T14\rtimes C_{3}$ |
$10T7\simeq A_{5}$ | $V_{4}\leq G_{v}$
$10T26\simeq{\rm PSL}_{2}(\mathbbm{F}_{9})$ | $D_{4}\leq G_{v}$
$10T32\simeq S_{6}$ |  (i) $V_{4}\leq G_{v}$ where $N_{\widetilde{G}}(V_{4})\simeq C_{8}\rtimes(C_{2}\times C_{2})$ for the normalizer $N_{\widetilde{G}}(V_{4})$ of $V_{4}$ in $\widetilde{G}$ with the normalizer $\widetilde{G}=N_{S_{10}}(G)\simeq{\rm Aut}(G)$ of $G$ in $S_{10}$ $($equivalently, $|{\rm Orb}_{V_{4}}(i)|=2$ for any $1\leq i\leq 10$$)$ or (ii) $D_{4}\leq G_{v}$ where $D_{4}\leq[G,G]\simeq A_{6}$
$14T30\simeq{\rm PSL}_{2}(\mathbbm{F}_{13})$ | $V_{4}\leq G_{v}$
$15T9\simeq(C_{5})^{2}\rtimes C_{3}$ | $(C_{5})^{2}\leq G_{v}$
$15T14\simeq(C_{5})^{2}\rtimes S_{3}$

We organize this paper as follows. In Section 2, we prepare some basic
definitions and known results about the rationality problem for norm one tori.
In Section 3, we recall our basic tool “flabby resolution of $G$-lattices” to
investigate algebraic $k$-tori. In Section 4, we give the proof of Theorem 1.5
and Theorem 1.6. In Section 5, the proofs of Theorem 1.15 and Theorem 1.17 are
given. In Section 6, we prove Theorem 1.18 by using Drakokhrust and Platonov’s
method for the Hasse norm principle for $K/k$. In Section 7, we will give an
application of Theorem 1.5, Theorem 1.6 and Theorem 1.15 to obtain the group
$T(k)/R$ of $R$-equivalence classes over a local field $k$ via the formula of
Colliot-Thélène and Sansuc. In Section 8, we also give an application of
Theorem 1.5, Theorem 1.6 and Theorem 1.18 to evaluate the Tamagawa number
$\tau(T)$ over a number field $k$ via Ono’s formula. In Section 9, we will
give GAP computations of $H^{1}(G,J_{G/H})$ as the appendix of this paper. GAP
algorithms will be given in Section 10 which are also available from
https://www.math.kyoto-u.ac.jp/~yamasaki/Algorithm/Norm1ToriHNP.

######  Acknowledgments.

We would like to thank Ming-chang Kang, Shizuo Endo and Boris Kunyavskii for
giving us useful and valuable comments. We also thank the referees for very
careful reading of the manuscript. This paper is greatly improved by their
helpful comments and suggestions. In particular, one of them tell us the
recent papers Macedo and Newton [MN] and Macedo [Mac] as a convergence of
interests. It may be interesting to compare this paper with them.

##  2\. Rationality problem for norm one tori

Let $k$ be a field, $K/k$ be a separable field extension of degree $n$ and
$L/k$ be the Galois closure of $K/k$. Let $G={\rm Gal}(L/k)$ and $H={\rm
Gal}(L/K)$ with $[G:H]=n$. The Galois group $G$ may be regarded as a
transitive subgroup of the symmetric group $S_{n}$ of degree $n$ via an
injection $G\to S_{n}$ which is derived from the action of $G$ on the left
cosets $\\{g_{1}H,\ldots,g_{n}H\\}$ by $g(g_{i}H)=(gg_{i})H$ for any $g\in G$
and we may assume that $H$ is the stabilizer of one of the letters in $G$,
i.e. $L=k(\theta_{1},\ldots,\theta_{n})$ and $K=k(\theta_{i})$ for some $1\leq
i\leq n$. The norm one torus $R^{(1)}_{K/k}(\mathbbm{G}_{m})$ has the
Chevalley module $J_{G/H}$ as its character module where
$J_{G/H}=(I_{G/H})^{\circ}={\rm Hom}_{\mathbbm{Z}}(I_{G/H},\mathbbm{Z})$ is
the dual lattice of $I_{G/H}={\rm Ker}\,\varepsilon$ and
$\varepsilon:\mathbbm{Z}[G/H]\rightarrow\mathbbm{Z}$ is the augmentation map
(see [Vos98, Section 4.8]). We have the exact sequence

$\displaystyle 0\rightarrow\mathbbm{Z}\rightarrow\mathbbm{Z}[G/H]\rightarrow
J_{G/H}\rightarrow 0$

and rank $J_{G/H}=n-1$. Write $J_{G/H}=\oplus_{1\leq i\leq
n-1}\mathbbm{Z}u_{i}$. We define the action of $G$ on
$L(x_{1},\ldots,x_{n-1})$ by
$\sigma(x_{i})=\prod_{j=1}^{n}x_{j}^{a_{i,j}}(1\leq i\leq n)$ for any
$\sigma\in G$, when $\sigma(u_{i})=\sum_{j=1}^{n}a_{i,j}u_{j}$
$(a_{i,j}\in\mathbbm{Z})$. Then the invariant field
$L(x_{1},\ldots,x_{n-1})^{G}$ may be identified with the function field of the
norm one torus $R^{(1)}_{K/k}(\mathbbm{G}_{m})$ (see [EM75, Section 1]).

Let $T=R^{(1)}_{K/k}(\mathbbm{G}_{m})$ be the norm one torus of $K/k$. The
rationality problem for norm one tori is investigated by [EM75], [CTS77],
[Hür84], [CTS87], [LeB95], [CK00], [LL00], [Flo], [End11], [HY17], [HY21],
[HHY20].

######  Theorem 2.1 (Colliot-Thélène and Sansuc [CTS87, Proposition 9.1],
[LeB95, Theorem 3.1], [CK00, Proposition 0.2], [LL00], Endo [End11, Theorem
4.1], see also [End11, Remark 4.2 and Theorem 4.3]).

Let $K/k$ be a non-Galois separable field extension of degree $n$ and $L/k$ be
the Galois closure of $K/k$. Assume that ${\rm Gal}(L/k)=S_{n}$, $n\geq 3$,
and ${\rm Gal}(L/K)=S_{n-1}$ is the stabilizer of one of the letters in
$S_{n}$. Then we have:
(i) $R^{(1)}_{K/k}(\mathbbm{G}_{m})$ is retract $k$-rational if and only if
$n$ is a prime;
(ii) $R^{(1)}_{K/k}(\mathbbm{G}_{m})$ is $($stably$)$ $k$-rational if and only
if $n=3$.

######  Theorem 2.2 (Endo [End11, Theorem 4.4], Hoshi and Yamasaki [HY17,
Corollary 1.11]).

Let $K/k$ be a non-Galois separable field extension of degree $n$ and $L/k$ be
the Galois closure of $K/k$. Assume that ${\rm Gal}(L/k)=A_{n}$, $n\geq 4$,
and ${\rm Gal}(L/K)=A_{n-1}$ is the stabilizer of one of the letters in
$A_{n}$. Then we have:
(i) $R^{(1)}_{K/k}(\mathbbm{G}_{m})$ is retract $k$-rational if and only if
$n$ is a prime.
(ii) $R^{(1)}_{K/k}(\mathbbm{G}_{m})$ is stably $k$-rational if and only if
$n=5$.

A necessary and sufficient condition for the classification of stably/retract
rational norm one tori $T=R^{(1)}_{K/k}(\mathbbm{G}_{m})$ with $[K:k]=n\leq
15$, but with one exception $G=9T27\simeq{\rm PSL}_{2}(\mathbbm{F}_{8})$ for
the stable rationality, was given in Hoshi and Yamasaki [HY21] (for the case
$n$ is a prime number or the case $n\leq 10$) and Hasegawa, Hoshi and Yamasaki
[HHY20] (for $n=12,14,15$).

##  3\. Strategy: flabby resolution of $G$-lattices

We recall some basic facts of the theory of flabby (flasque) $G$-lattices (see
[CTS77], [Swa83], [Vos98, Chapter 2], [Lor05, Chapter 2], [Swa10]). Recall
also that we may take the $G$-lattice $\widehat{T}$ for an algebraic $k$-torus
$T$ (see Section 1).

######  Definition 3.1.

Let $G$ be a finite group and $M$ be a $G$-lattice (i.e. finitely generated
$\mathbbm{Z}[G]$-module which is $\mathbbm{Z}$-free as an abelian group).
(i) $M$ is called a permutation $G$-lattice if $M$ has a $\mathbbm{Z}$-basis
permuted by $G$, i.e. $M\simeq\oplus_{1\leq i\leq m}\mathbbm{Z}[G/H_{i}]$ for
some subgroups $H_{1},\ldots,H_{m}$ of $G$.
(ii) $M$ is called a stably permutation $G$-lattice if $M\oplus P\simeq
P^{\prime}$ for some permutation $G$-lattices $P$ and $P^{\prime}$.
(iii) $M$ is called invertible (or permutation projective) if it is a direct
summand of a permutation $G$-lattice, i.e. $P\simeq M\oplus M^{\prime}$ for
some permutation $G$-lattice $P$ and a $G$-lattice $M^{\prime}$.
(iv) $M$ is called flabby (or flasque) if $\widehat{H}^{-1}(H,M)=0$ for any
subgroup $H$ of $G$ where $\widehat{H}$ is the Tate cohomology.
(v) $M$ is called coflabby (or coflasque) if $H^{1}(H,M)=0$ for any subgroup
$H$ of $G$.

######  Definition 3.2 (see [EM75, Section 1], [Vos98, Section 4.7]).

Let $\mathcal{C}(G)$ be the category of all $G$-lattices. Let $\mathcal{S}(G)$
be the full subcategory of $\mathcal{C}(G)$ of all permutation $G$-lattices
and $\mathcal{D}(G)$ be the full subcategory of $\mathcal{C}(G)$ of all
invertible $G$-lattices. Let

$\displaystyle\mathcal{H}^{i}(G)=\\{M\in\mathcal{C}(G)\mid\widehat{H}^{i}(H,M)=0\
{\rm for\ any}\ H\leq G\\}\ (i=\pm 1)$

be the class of “$\widehat{H}^{i}$-vanish” $G$-lattices where
$\widehat{H}^{i}$ is the Tate cohomology. Then we have the inclusions
$\mathcal{S}(G)\subset\mathcal{D}(G)\subset\mathcal{H}^{i}(G)\subset\mathcal{C}(G)$
$(i=\pm 1)$.

######  Definition 3.3.

We say that two $G$-lattices $M_{1}$ and $M_{2}$ are similar if there exist
permutation $G$-lattices $P_{1}$ and $P_{2}$ such that $M_{1}\oplus
P_{1}\simeq M_{2}\oplus P_{2}$. We denote the similarity class of $M$ by
$[M]$. The set of similarity classes $\mathcal{C}(G)/\mathcal{S}(G)$ becomes a
commutative monoid (with respect to the sum $[M_{1}]+[M_{2}]:=[M_{1}\oplus
M_{2}]$ and the zero $0=[P]$ where $P\in\mathcal{S}(G)$).

######  Theorem 3.4 (Endo and Miyata [EM75, Lemma 1.1], Colliot-Thélène and
Sansuc [CTS77, Lemma 3], see also [Swa83, Lemma 8.5], [Lor05, Lemma 2.6.1]).

For any $G$-lattice $M$, there exists a short exact sequence of $G$-lattices
$0\rightarrow M\rightarrow P\rightarrow F\rightarrow 0$ where $P$ is
permutation and $F$ is flabby.

######  Definition 3.5.

The exact sequence $0\rightarrow M\rightarrow P\rightarrow F\rightarrow 0$ as
in Theorem 3.4 is called a flabby resolution of the $G$-lattice $M$.
$\rho_{G}(M)=[F]\in\mathcal{C}(G)/\mathcal{S}(G)$ is called the flabby class
of $M$, denoted by $[M]^{fl}=[F]$. Note that $[M]^{fl}$ is well-defined: if
$[M]=[M^{\prime}]$, $[M]^{fl}=[F]$ and $[M^{\prime}]^{fl}=[F^{\prime}]$ then
$F\oplus P_{1}\simeq F^{\prime}\oplus P_{2}$ for some permutation $G$-lattices
$P_{1}$ and $P_{2}$, and therefore $[F]=[F^{\prime}]$ (cf. [Swa83, Lemma
8.7]). We say that $[M]^{fl}$ is invertible if $[M]^{fl}=[E]$ for some
invertible $G$-lattice $E$.

For $G$-lattice $M$, it is not difficult to see

$\displaystyle\textrm{permutation}\ \ \Rightarrow\ \ $
$\displaystyle\textrm{stably\ permutation}\ \ \Rightarrow\ \
\textrm{invertible}\ \ \Rightarrow\ \ \textrm{flabby\ and\ coflabby}$
$\displaystyle\hskip 22.76219pt\Downarrow\hskip 96.73936pt\Downarrow$
$\displaystyle\hskip 19.91692pt[M]^{fl}=0\hskip 28.45274pt\Rightarrow\hskip
14.22636pt[M]^{fl}\ \textrm{is\ invertible}.$

The above implications in each step cannot be reversed (see, for example,
[HY17, Section 1]).

Let $T$ be an algebraic $k$-torus and $\widehat{T}={\rm
Hom}(T,\mathbbm{G}_{m})$ be the character module of $T$. Then $\widehat{T}$
becomes a $G$-lattice where $G={\rm Gal}(L/k)$ is the Galois group of $L/k$
and $L$ is the minimal splitting field of $T$. The flabby class
$\rho_{G}(\widehat{T})=[\widehat{T}]^{fl}$ plays crucial role in the
rationality problem for $T$ as follows (see Voskresenskii’s fundamental book
[Vos98, Section 4.6] and Kunyavskii [Kun07], see also e.g. Swan [Swa83],
Kunyavskii [Kun90, Section 2], Lemire, Popov and Reichstein [LPR06, Section
2], Kang [Kan12], Yamasaki [Yam12], Hoshi and Yamasaki [HY17]):

######  Theorem 3.6 (Endo and Miyata [EM73], Voskresenskii [Vos74], Saltman
[Sal84]).

Let $T$ and $T^{\prime}$ be algebraic $k$-tori with the same minimal splitting
field $L$. Then we have:
(i) $($Endo and Miyata [EM73, Theorem 1.6]$)$ $[\widehat{T}]^{fl}=0$ if and
only if $T$ is stably $k$-rational;
(ii) $($Voskresenskii [Vos74, Theorem 2]$)$
$[\widehat{T}]^{fl}=[\widehat{T}^{\prime}]^{fl}$ if and only if $T$ and
$T^{\prime}$ are stably $k$-isomorphic;
(iii) $($Saltman [Sal84, Theorem 3.14]$)$ $[\widehat{T}]^{fl}$ is invertible
if and only if $T$ is retract $k$-rational.

For norm one tori $T=R^{(1)}_{K/k}(\mathbbm{G}_{m})$, recall that
$\widehat{T}=J_{G/H}$. Hence we have

$\displaystyle[J_{G/H}]^{fl}=0\,\ \ \Rightarrow\ \ [J_{G/H}]^{fl}\ \textrm{is\
invertible}\ \ \Rightarrow\ \ H^{1}(G,[J_{G/H}]^{fl})=0\,\ \ \Rightarrow\ \
A(T)=0\ \textrm{and}\ \Sha(T)=0$

where the last implication holds over a global field $k$ (see also Colliot-
Thélène and Sansuc [CTS77, page 29]). The last conditions mean that $T$ has
the weak approximation property and the Hasse norm principle holds for $K/k$
(see Section 1). In particular, it follows from Theorem 2.1 that
$H^{1}(G,[J_{G/H}]^{fl})=0$ and hence $A(T)=0$ and $\Sha(T)=0$ when $G=pTm\leq
S_{p}$ is a transitive subgroup of $S_{p}$ of prime degree and $H\leq G$ with
$[G:H]=p$ (see also the first paragraph of Section 5).

##  4\. Proof of Theorem 1.5 and Theorem 1.6

We will give the proof of Theorem 4.1 and Theorem 4.2 which are detailed
statements of Theorem 1.5 and Theorem 1.6 respectively:

######  Theorem 4.1.

Let $k$ be a field, $T$ be an algebraic $k$-torus of dimension $4$ and $X$ be
a smooth $k$-compactification of $T$. Among $216$ cases of not retract
rational algebraic $k$-tori $T$, there exist exactly $2$ $($resp. $20$,
$194$$)$ algebraic $k$-tori with $H^{1}(k,{\rm
Pic}\,\overline{X})\simeq(\mathbbm{Z}/2\mathbbm{Z})^{\oplus 2}$ $($resp.
$H^{1}(k,{\rm Pic}\,\overline{X})\simeq\mathbbm{Z}/2\mathbbm{Z}$,
$H^{1}(k,{\rm Pic}\,\overline{X})=0$$)$. Moreover, for the character module
$\widehat{T}\simeq M_{G}$ of $T$ with $H^{1}(k,{\rm Pic}\,\overline{X})\simeq
H^{1}(G,[M_{G}]^{fl})$, we have
(i) $H^{1}(G,[M_{G}]^{fl})\simeq(\mathbbm{Z}/2\mathbbm{Z})^{\oplus 2}$ if and
only if the GAP ID of $G$ is one of $(4,32,1,2)$ and $(4,33,3,1)$ where
$M_{G}$ is an indecomposable $G$-lattice of rank $4$ and $G$ is isomorphic to
$Q_{8}$ and ${\rm SL}_{2}(\mathbbm{F}_{3})$ respectively;
(ii) $H^{1}(G,[M_{G}]^{fl})\simeq\mathbbm{Z}/2\mathbbm{Z}$ if and only if
(ii-1) the GAP ID of $G$ is one of $(4,5,1,12)$, $(4,5,2,8)$, $(4,6,2,10)$,
$(4,12,2,6)$, $(4,12,4,12)$, $(4,12,5,10)$, $(4,18,1,3)$, $(4,18,4,4)$,
$(4,32,2,2)$, $(4,32,3,2)$, $(4,32,4,2)$, $(4,32,6,2)$, $(4,33,5,1)$,
$(4,33,6,1)$, $(4,33,9,1)$ where $M_{G}$ is an indecomposable $G$-lattice of
rank $4$ and $G$ is isomorphic to $V_{4}$, $(C_{2})^{3}$, $(C_{2})^{3}$,
$C_{4}\times C_{2}$, $D_{4}$, $C_{2}\times D_{4}$, $C_{4}\times C_{2}$,
$C_{2}\times D_{4}$, $Q_{16}$, $QD_{8}$, $(C_{4}\times C_{2})\rtimes C_{2}$,
$C_{8}\rtimes V_{4}$, $((C_{4}\times C_{2})\rtimes C_{2})\rtimes C_{3}$, ${\rm
GL}_{2}(\mathbbm{F}_{3})$, ${\rm GL}_{2}(\mathbbm{F}_{3})\rtimes C_{2}$
respectively; or
(ii-2) the GAP ID of $G$ is one of $(4,4,3,6)$, $(4,5,1,9)$, $(4,6,2,9)$,
$(4,24,1,5)$, $(4,25,2,4)$ where $M_{G}$ is a decomposable $G$-lattice of rank
$4=3+1$ and $G$ is isomorphic to $V_{4}$, $V_{4}$, $(C_{2})^{3}$, $A_{4}$,
$C_{2}\times A_{4}$ respectively.

###### Proof.

It follows from [HY17, Theorem 1.9] that among $710$ cases of $4$-dimensional
algebraic $k$-tori, there exist exactly $216$ algebraic $k$-tori which are not
retract $k$-rational. Because if $T$ is retract $k$-rational, then
$H^{1}(k,{\rm Pic}\,\overline{X})\simeq H^{1}(G,[M_{G}]^{fl})=0$, we should
check only the $216$ cases. The GAP IDs of such $216$ groups $G\leq{\rm
GL}_{4}(\mathbbm{Z})$ with $[M_{G}]^{fl}$ is not invertible, are given in
[HY17, Tables 3, 4] (see [HY17, Chapter 3] for the explanation of GAP ID).
They are also given in [HY17, Example 10.1] as the lists N4 (resp. N31) when
$M_{G}$ is indecomposable (resp. decomposable with rank $4=3+1$) and available
from
https://www.math.kyoto-u.ac.jp/~yamasaki/Algorithm/MultInvField/NonInv.dat.

Then we apply the function FlabbyResolutionLowRank($G$).actionF in [HHY20,
Algorithm 4.1] which returns a suitable flabby class $F=[M_{G}]^{fl}$ with low
rank by using the backtracking techniques. The function H1 may compute the
group $H^{1}(G,F)$ (see Example 4.4). The related functions are available from
https://www.math.kyoto-u.ac.jp/~yamasaki/Algorithm/RatProbNorm1Tori/. ∎

######  Theorem 4.2.

Let $k$ be a field, $T$ be an algebraic $k$-torus of dimension $5$ and $X$ be
a smooth $k$-compactification of $T$. Among $3003$ cases of not retract
rational algebraic $k$-tori $T$, there exist exactly $11$ $($resp. $263$,
$2729$$)$ algebraic $k$-tori with $H^{1}(k,{\rm
Pic}\,\overline{X})\simeq(\mathbbm{Z}/2\mathbbm{Z})^{\oplus 2}$ $($resp.
$H^{1}(k,{\rm Pic}\,\overline{X})\simeq\mathbbm{Z}/2\mathbbm{Z}$,
$H^{1}(k,{\rm Pic}\,\overline{X})=0$$)$. Moreover, for the character module
$\widehat{T}\simeq M_{G}$ of $T$ with $H^{1}(k,{\rm Pic}\,\overline{X})\simeq
H^{1}(G,[M_{G}]^{fl})$, we have
(i) $H^{1}(G,[M_{G}]^{fl})\simeq(\mathbbm{Z}/2\mathbbm{Z})^{\oplus 2}$ if and
only if
(i-1) the CARAT ID of $G$ is one of the $6$ triples $(5,31,26)$, $(5,31,27)$,
$(5,664,2)$, $(5,669,2)$, $(5,670,2)$, $(5,773,4)$ where $M_{G}$ is an
indecomposable $G$-lattice of rank $5$ and $G$ is isomorphic to $(C_{2})^{3}$,
$(C_{2})^{3}$, $C_{2}\times Q_{8}$, $(C_{4}\times C_{2})\rtimes C_{2}$,
$(C_{4}\times C_{2})\rtimes C_{2}$, $Q_{8}$ respectively; or
(i-2) the CARAT ID of $G$ is one of the $5$ triples $(5,664,1)$, $(5,773,3)$,
$(5,774,3)$, $(5,691,1)$, $(5,730,1)$ where $M_{G}$ is a decomposable
$G$-lattice of rank $5=4+1$ and $G$ is isomorphic to $C_{2}\times Q_{8}$,
$Q_{8}$, $Q_{8}$, ${\rm SL}_{2}(\mathbbm{F}_{3})$, $C_{2}\times{\rm
SL}_{2}(\mathbbm{F}_{3})$ respectively.
(ii) $H^{1}(G,[M_{G}]^{fl})\simeq\mathbbm{Z}/2\mathbbm{Z}$ if and only if
(ii-1) the CARAT ID of $G$ is one of the $141$ triples as in Example 4.5 where
$M_{G}$ is an indecomposable $G$-lattice of rank $5$;
(ii-2) the CARAT ID of $G$ is one of the $73$ triples as in Example 4.5 where
$M_{G}$ is a decomposable $G$-lattice of rank $5=4+1$;
(ii-3) the CARAT ID of $G$ is one of the $36$ triples as in Example 4.5 where
$M_{G}$ is a decomposable $G$-lattice of rank $5=3+2$; or
(ii-4) the CARAT ID of $G$ is one of the $13$ triples as in Example 4.5 where
$M_{G}$ is a decomposable $G$-lattice of rank $5=3+1+1$.

###### Proof.

The method is the same as in the proof of Theorem 4.1. By [HY17, Theorem
1.12], among $6079$ cases of $5$-dimensional algebraic $k$-tori, there exist
exactly $3003$ algebraic $k$-tori which are not retract $k$-rational. The
CARAT IDs of such $3003$ groups $G\leq{\rm GL}_{5}(\mathbbm{Z})$ with
$[M_{G}]^{fl}$ is not invertible, are given in [HY17, Tables 12, 13, 14, 15].
They are also given in [HY17, Example 4.12 and Example 11.1] as the lists N5,
N41, N32, N311 when $M_{G}$ is indecomposable (resp. decomposable with rank
$5=4+1$, $5=3+1$, $5=3+1+1$) and available from
https://www.math.kyoto-u.ac.jp/~yamasaki/Algorithm/MultInvField/NonInv5.dat.

Then we apply the functions FlabbyResolutionLowRank($G$).actionF in [HHY20,
Algorithm 4.1] and H1 to get $H^{1}(G,[M_{G}]^{fl})$ (see Example 4.5 and also
the proof of Theorem 4.1). ∎

######  Example 4.3 (Classification of $H^{1}(G,[M_{G}]^{fl})\neq 0$ for
$G\leq{\rm GL}_{3}(\mathbbm{Z})$).

    gap> Read("FlabbyResolutionFromBase.gap");
    gap> Read("NonInv.dat");
    # N3 is the list of GAP IDs (Crystcat IDs) of indecomposable lattice of rank 3
    # whose flabby class [M_G]^fl is not invertible [HY17, Example 4.12]
    gap> N3;
    [ [ 3, 3, 1, 3 ], [ 3, 3, 3, 3 ], [ 3, 3, 3, 4 ], [ 3, 4, 3, 2 ], [ 3, 4, 4, 2 ],
      [ 3, 4, 6, 3 ], [ 3, 4, 7, 2 ], [ 3, 7, 1, 2 ], [ 3, 7, 2, 2 ], [ 3, 7, 2, 3 ],
      [ 3, 7, 3, 2 ], [ 3, 7, 3, 3 ], [ 3, 7, 4, 2 ], [ 3, 7, 5, 2 ], [ 3, 7, 5, 3 ] ]
    gap> Length(N3); # there exist 15 not retract rational tori in dim=3 [HY17, Table 1]
    15
    gap> N3g:=List(N3,x->MatGroupZClass(x[1],x[2],x[3],x[4]));;
    gap> List(N3g,StructureDescription);
    [ "C2 x C2", "C2 x C2 x C2", "C2 x C2 x C2", "C4 x C2", "D8", "D8", "C2 x D8",
    "A4", "C2 x A4", "C2 x A4", "S4", "S4", "S4", "C2 x S4", "C2 x S4" ]
    gap> N3gF:=List(N3g,x->FlabbyResolutionLowRank(x).actionF);;
    gap> N3H1F:=List(N3gF,x->Filtered(H1(x),y->y>1)); # H1(F)
    [ [ 2 ], [  ], [  ], [  ], [  ], [  ], [  ], [ 2 ], [  ], [  ], [  ], [  ], [  ], [  ], [  ] ]
    gap> N3H1FC2:=Filtered([1..Length(N3gF)],x->N3H1F[x]=[2]);
    [ 1, 8 ]
    gap> List(N3H1FC2,x->N3[x]); # GAP ID’s of F with H1(F)=C2
    [ [ 3, 3, 1, 3 ], [ 3, 7, 1, 2 ] ]
    gap> List(N3H1FC2,x->StructureDescription(N3g[x]));
    [ "C2 x C2", "A4" ]

######  Example 4.4 (Classification of $H^{1}(G,[M_{G}]^{fl})\neq 0$ for
$G\leq{\rm GL}_{4}(\mathbbm{Z})$).

    gap> Read("FlabbyResolutionFromBase.gap");
    gap> Read("NonInv.dat");
    # N4 is the list of GAP IDs (Crystcat IDs) of indecomposable lattice of rank 4
    # whose flabby class [M_G]^fl is not invertible [HY17, Example 4.12]
    gap> N4;
    [ [ 4, 5, 1, 12 ], [ 4, 5, 2, 5 ], [ 4, 5, 2, 8 ], [ 4, 5, 2, 9 ], [ 4, 6, 1, 6 ],
      [ 4, 6, 1, 11 ], [ 4, 6, 2, 6 ], [ 4, 6, 2, 10 ], [ 4, 6, 2, 12 ], [ 4, 6, 3, 4 ],
      [ 4, 6, 3, 7 ], [ 4, 6, 3, 8 ], [ 4, 12, 2, 5 ], [ 4, 12, 2, 6 ], [ 4, 12, 3, 11 ],
      [ 4, 12, 4, 10 ], [ 4, 12, 4, 11 ], [ 4, 12, 4, 12 ], [ 4, 12, 5, 8 ], [ 4, 12, 5, 9 ],
      [ 4, 12, 5, 10 ], [ 4, 12, 5, 11 ], [ 4, 13, 1, 5 ], [ 4, 13, 2, 5 ], [ 4, 13, 3, 5 ],
      [ 4, 13, 4, 5 ], [ 4, 13, 5, 4 ], [ 4, 13, 5, 5 ], [ 4, 13, 6, 5 ], [ 4, 13, 7, 9 ],
      [ 4, 13, 7, 10 ], [ 4, 13, 7, 11 ], [ 4, 13, 8, 5 ], [ 4, 13, 8, 6 ], [ 4, 13, 9, 4 ],
      [ 4, 13, 9, 5 ], [ 4, 13, 10, 4 ], [ 4, 13, 10, 5 ], [ 4, 18, 1, 3 ], [ 4, 18, 2, 4 ],
      [ 4, 18, 2, 5 ], [ 4, 18, 3, 5 ], [ 4, 18, 3, 6 ], [ 4, 18, 3, 7 ], [ 4, 18, 4, 4 ],
      [ 4, 18, 4, 5 ], [ 4, 18, 5, 5 ], [ 4, 18, 5, 6 ], [ 4, 18, 5, 7 ], [ 4, 19, 1, 2 ],
      [ 4, 19, 2, 2 ], [ 4, 19, 3, 2 ], [ 4, 19, 4, 3 ], [ 4, 19, 4, 4 ], [ 4, 19, 5, 2 ],
      [ 4, 19, 6, 2 ], [ 4, 22, 1, 1 ], [ 4, 22, 2, 1 ], [ 4, 22, 3, 1 ], [ 4, 22, 4, 1 ],
      [ 4, 22, 5, 1 ], [ 4, 22, 5, 2 ], [ 4, 22, 6, 1 ], [ 4, 22, 7, 1 ], [ 4, 22, 8, 1 ],
      [ 4, 22, 9, 1 ], [ 4, 22, 10, 1 ], [ 4, 22, 11, 1 ], [ 4, 24, 2, 4 ], [ 4, 24, 2, 6 ],
      [ 4, 24, 4, 4 ], [ 4, 24, 5, 4 ], [ 4, 24, 5, 6 ], [ 4, 25, 1, 3 ], [ 4, 25, 2, 3 ],
      [ 4, 25, 2, 5 ], [ 4, 25, 3, 3 ], [ 4, 25, 4, 3 ], [ 4, 25, 5, 3 ], [ 4, 25, 5, 5 ],
      [ 4, 25, 6, 3 ], [ 4, 25, 6, 5 ], [ 4, 25, 7, 3 ], [ 4, 25, 8, 3 ], [ 4, 25, 9, 3 ],
      [ 4, 25, 9, 5 ], [ 4, 25, 10, 3 ], [ 4, 25, 10, 5 ], [ 4, 25, 11, 3 ], [ 4, 25, 11, 5 ],
      [ 4, 29, 1, 1 ], [ 4, 29, 1, 2 ], [ 4, 29, 2, 1 ], [ 4, 29, 3, 1 ], [ 4, 29, 3, 2 ],
      [ 4, 29, 3, 3 ], [ 4, 29, 4, 1 ], [ 4, 29, 4, 2 ], [ 4, 29, 5, 1 ], [ 4, 29, 6, 1 ],
      [ 4, 29, 7, 1 ], [ 4, 29, 7, 2 ], [ 4, 29, 8, 1 ], [ 4, 29, 8, 2 ], [ 4, 29, 9, 1 ],
      [ 4, 32, 1, 2 ], [ 4, 32, 2, 2 ], [ 4, 32, 3, 2 ], [ 4, 32, 4, 2 ], [ 4, 32, 5, 2 ],
      [ 4, 32, 5, 3 ], [ 4, 32, 6, 2 ], [ 4, 32, 7, 2 ], [ 4, 32, 8, 2 ], [ 4, 32, 9, 4 ],
      [ 4, 32, 9, 5 ], [ 4, 32, 10, 2 ], [ 4, 32, 11, 2 ], [ 4, 32, 11, 3 ], [ 4, 32, 12, 2 ],
      [ 4, 32, 13, 3 ], [ 4, 32, 13, 4 ], [ 4, 32, 14, 3 ], [ 4, 32, 14, 4 ], [ 4, 32, 15, 2 ],
      [ 4, 32, 16, 2 ], [ 4, 32, 16, 3 ], [ 4, 32, 17, 2 ], [ 4, 32, 18, 2 ], [ 4, 32, 18, 3 ],
      [ 4, 32, 19, 2 ], [ 4, 32, 19, 3 ], [ 4, 32, 20, 2 ], [ 4, 32, 20, 3 ], [ 4, 32, 21, 2 ],
      [ 4, 32, 21, 3 ], [ 4, 33, 1, 1 ], [ 4, 33, 3, 1 ], [ 4, 33, 4, 1 ], [ 4, 33, 5, 1 ],
      [ 4, 33, 6, 1 ], [ 4, 33, 7, 1 ], [ 4, 33, 8, 1 ], [ 4, 33, 9, 1 ], [ 4, 33, 10, 1 ],
      [ 4, 33, 11, 1 ], [ 4, 33, 12, 1 ], [ 4, 33, 13, 1 ], [ 4, 33, 14, 1 ], [ 4, 33, 14, 2 ],
      [ 4, 33, 15, 1 ], [ 4, 33, 16, 1 ] ]
    gap> Length(N4); # there exist 152 not retract rational tori in dim=4 [HY17, Table 4]
    152
    gap> N4g:=List(N4,x->MatGroupZClass(x[1],x[2],x[3],x[4]));
    gap> N4gF:=List(N4g,x->FlabbyResolutionLowRank(x).actionF);;
    gap> N4H1F:=List(N4gF,x->Filtered(H1(x),y->y>1));;
    gap> Collected(N4H1F);
    [ [ [  ], 135 ], [ [ 2 ], 15 ], [ [ 2, 2 ], 2 ] ]

    gap> N4H1FC2xC2:=Filtered([1..Length(N4H1F)],x->N4H1F[x]=[2,2]);
    [ 106, 138 ]
    gap> List(N4H1FC2xC2,x->N4[x]); # GAP ID’s of F with H1(F)=C2xC2
    [ [ 4, 32, 1, 2 ], [ 4, 33, 3, 1 ] ]
    gap> List(N4H1FC2xC2,x->StructureDescription(N4g[x]));
    [ "Q8", "SL(2,3)" ]

    gap> N4H1FC2:=Filtered([1..Length(N4H1F)],x->N4H1F[x]=[2]);
    [ 1, 3, 8, 14, 18, 21, 39, 45, 107, 108, 109, 112, 140, 141, 144 ]
    gap> List(N4H1FC2,x->N4[x]); # GAP ID’s of F with H1(F)=C2
    [ [ 4, 5, 1, 12 ], [ 4, 5, 2, 8 ], [ 4, 6, 2, 10 ], [ 4, 12, 2, 6 ], [ 4, 12, 4, 12 ],
      [ 4, 12, 5, 10 ], [ 4, 18, 1, 3 ], [ 4, 18, 4, 4 ], [ 4, 32, 2, 2 ], [ 4, 32, 3, 2 ],
      [ 4, 32, 4, 2 ], [ 4, 32, 6, 2 ], [ 4, 33, 5, 1 ], [ 4, 33, 6, 1 ], [ 4, 33, 9, 1 ] ]
    gap> List(N4H1FC2,x->StructureDescription(N4g[x]));
    [ "C2 x C2", "C2 x C2 x C2", "C2 x C2 x C2", "C4 x C2", "D8", "C2 x D8", "C4 x C2",
      "C2 x D8", "C8 : C2", "QD16", "(C4 x C2) : C2", "C8 : (C2 x C2)",
      "((C4 x C2) : C2) : C3", "GL(2,3)", "GL(2,3) : C2" ]

    # N31 is the list of GAP IDs (Crystcat IDs) of decomposable lattice of rank 4=3+1
    # whose flabby class [M_G]^fl is not invertible [HY17, Example 4.12]
    gap> N31;
    [ [ 4, 4, 3, 6 ], [ 4, 4, 4, 4 ], [ 4, 4, 4, 6 ], [ 4, 5, 1, 9 ], [ 4, 5, 2, 4 ],
      [ 4, 5, 2, 7 ], [ 4, 6, 1, 4 ], [ 4, 6, 1, 8 ], [ 4, 6, 2, 4 ], [ 4, 6, 2, 8 ],
      [ 4, 6, 2, 9 ], [ 4, 6, 3, 3 ], [ 4, 6, 3, 6 ], [ 4, 7, 3, 2 ], [ 4, 7, 4, 3 ],
      [ 4, 7, 5, 2 ], [ 4, 7, 7, 2 ], [ 4, 12, 2, 4 ], [ 4, 12, 3, 7 ], [ 4, 12, 4, 6 ],
      [ 4, 12, 4, 8 ], [ 4, 12, 4, 9 ], [ 4, 12, 5, 6 ], [ 4, 12, 5, 7 ], [ 4, 13, 1, 3 ],
      [ 4, 13, 2, 4 ], [ 4, 13, 3, 4 ], [ 4, 13, 4, 3 ], [ 4, 13, 5, 3 ], [ 4, 13, 6, 3 ],
      [ 4, 13, 7, 6 ], [ 4, 13, 7, 7 ], [ 4, 13, 7, 8 ], [ 4, 13, 8, 3 ], [ 4, 13, 8, 4 ],
      [ 4, 13, 9, 3 ], [ 4, 13, 10, 3 ], [ 4, 24, 1, 5 ], [ 4, 24, 2, 3 ], [ 4, 24, 2, 5 ],
      [ 4, 24, 3, 5 ], [ 4, 24, 4, 3 ], [ 4, 24, 4, 5 ], [ 4, 24, 5, 3 ], [ 4, 24, 5, 5 ],
      [ 4, 25, 1, 2 ], [ 4, 25, 1, 4 ], [ 4, 25, 2, 4 ], [ 4, 25, 3, 2 ], [ 4, 25, 3, 4 ],
      [ 4, 25, 4, 4 ], [ 4, 25, 5, 2 ], [ 4, 25, 5, 4 ], [ 4, 25, 6, 2 ], [ 4, 25, 6, 4 ],
      [ 4, 25, 7, 2 ], [ 4, 25, 7, 4 ], [ 4, 25, 8, 2 ], [ 4, 25, 8, 4 ], [ 4, 25, 9, 4 ],
      [ 4, 25, 10, 2 ], [ 4, 25, 10, 4 ], [ 4, 25, 11, 2 ], [ 4, 25, 11, 4 ] ]
    gap> Length(N31); # there exist 64 not retract rational tori in dim=4=3+1 [HY17, Table 3]
    64
    gap> N31g:=List(N31,x->MatGroupZClass(x[1],x[2],x[3],x[4]));;
    gap> N31gF:=List(N31g,x->FlabbyResolutionLowRank(x).actionF);;
    gap> N31H1F:=List(N31gF,x->Filtered(H1(x),y->y>1));;
    gap> Collected(N31H1F);
    [ [ [  ], 59 ], [ [ 2 ], 5 ] ]
    gap> N31H1FC2:=Filtered([1..Length(N31H1F)],x->N31H1F[x]=[2]);
    [ 1, 4, 11, 38, 48 ]

    gap> List(N31H1FC2,x->N31[x]); # GAP ID’s of F with H1(F)=C2
    [ [ 4, 4, 3, 6 ], [ 4, 5, 1, 9 ], [ 4, 6, 2, 9 ], [ 4, 24, 1, 5 ], [ 4, 25, 2, 4 ] ]
    gap> List(N31H1FC2,x->StructureDescription(N31g[x]));
    [ "C2 x C2", "C2 x C2", "C2 x C2 x C2", "A4", "C2 x A4" ]

######  Example 4.5 (Classification of $H^{1}(G,[M_{G}]^{fl})\neq 0$ for
$G\leq{\rm GL}_{5}(\mathbbm{Z})$).

    gap> Read("FlabbyResolutionFromBase.gap");
    gap> Read("caratnumber.gap");
    gap> Read("NonInv5.dat");
    # N5 is the list of CARAT IDs of indecomposable lattice of rank 5
    # whose flabby class [M_G]^fl is not invertible [HY17, Example 4.12]
    gap> N5g:=List(N5,x->CaratMatGroupZClass(x[1],x[2],x[3]));;
    gap> Length(N5g); # there exist 1141 not retract rational tori in dim=5 [HY17, Table 15]
    1141
    gap> N5gF:=List(N5g,x->FlabbyResolutionLowRank(x).actionF);;
    gap> N5H1F:=List(N5gF,x->Filtered(H1(x),y->y>1));
    gap> Collected(N5H1F);
    [ [ [  ], 994 ], [ [ 2 ], 141 ], [ [ 2, 2 ], 6 ] ]

    gap> N5H1FC2xC2:=Filtered([1..Length(N5H1F)],x->N5H1F[x]=[2,2]);
    [ 69, 70, 906, 913, 915, 1064 ]
    gap> List(N5H1FC2xC2,x->N5[x]); # CARAT ID’s of F with H1(F)=C2xC2
    [ [ 5, 31, 26 ], [ 5, 31, 27 ], [ 5, 664, 2 ], [ 5, 669, 2 ], [ 5, 670, 2 ], [ 5, 773, 4 ] ]
    gap> List(N5H1FC2xC2,x->StructureDescription(N5g[x]));
    [ "C2 x C2 x C2", "C2 x C2 x C2", "C2 x Q8", "(C4 x C2) : C2", "(C4 x C2) : C2", "Q8" ]

    gap> N5H1FC2:=Filtered([1..Length(N5H1F)],x->N5H1F[x]=[2]);
    [ 3, 4, 5, 6, 8, 11, 19, 20, 27, 36, 37, 42, 43, 61, 63, 67, 71, 72, 74, 78, 79, 86, 88, 89,
      96, 99, 100, 103, 115, 116, 128, 129, 130, 131, 142, 143, 158, 159, 160, 173, 174, 178,
      179, 185, 186, 187, 188, 191, 193, 199, 200, 221, 222, 238, 242, 243, 253, 254, 288, 292,
      293, 316, 317, 318, 324, 327, 331, 333, 334, 337, 339, 348, 358, 362, 375, 376, 378, 389,
      401, 403, 404, 406, 407, 410, 414, 419, 423, 425, 440, 470, 480, 495, 511, 523, 540, 573,
      588, 590, 591, 592, 593, 595, 596, 597, 606, 680, 715, 723, 762, 852, 853, 854, 855, 908,
      909, 912, 916, 918, 921, 922, 948, 957, 961, 964, 970, 971, 973, 974, 976, 980, 982, 984,
      1037, 1060, 1065, 1114, 1115, 1116, 1117, 1129, 1130 ]
    gap> List(N5H1FC2,x->N5[x]); # CARAT ID’s of F with H1(F)=C2
    [ [ 5, 18, 23 ], [ 5, 19, 17 ], [ 5, 20, 14 ], [ 5, 20, 17 ], [ 5, 21, 17 ], [ 5, 24, 23 ],
      [ 5, 25, 27 ], [ 5, 25, 28 ], [ 5, 26, 21 ], [ 5, 26, 40 ], [ 5, 26, 41 ], [ 5, 27, 14 ],
      [ 5, 27, 15 ], [ 5, 30, 24 ], [ 5, 30, 28 ], [ 5, 31, 18 ], [ 5, 31, 31 ], [ 5, 31, 32 ],
      [ 5, 31, 36 ], [ 5, 31, 44 ], [ 5, 31, 45 ], [ 5, 32, 36 ], [ 5, 32, 44 ], [ 5, 32, 51 ],
      [ 5, 39, 5 ], [ 5, 71, 19 ], [ 5, 71, 22 ], [ 5, 71, 25 ], [ 5, 72, 34 ], [ 5, 72, 36 ],
      [ 5, 73, 32 ], [ 5, 73, 34 ], [ 5, 73, 36 ], [ 5, 73, 37 ], [ 5, 75, 34 ], [ 5, 75, 36 ],
      [ 5, 76, 49 ], [ 5, 76, 50 ], [ 5, 76, 51 ], [ 5, 78, 12 ], [ 5, 78, 15 ], [ 5, 78, 28 ],
      [ 5, 78, 31 ], [ 5, 79, 12 ], [ 5, 79, 15 ], [ 5, 79, 17 ], [ 5, 79, 18 ], [ 5, 79, 31 ],
      [ 5, 79, 36 ], [ 5, 80, 12 ], [ 5, 80, 15 ], [ 5, 83, 15 ], [ 5, 83, 17 ], [ 5, 86, 9 ],
      [ 5, 87, 9 ], [ 5, 87, 11 ], [ 5, 88, 34 ], [ 5, 88, 36 ], [ 5, 93, 9 ], [ 5, 94, 9 ],
      [ 5, 94, 11 ], [ 5, 99, 23 ], [ 5, 99, 24 ], [ 5, 99, 25 ], [ 5, 100, 12 ], [ 5, 100, 23 ],
      [ 5, 100, 28 ], [ 5, 101, 17 ], [ 5, 101, 18 ], [ 5, 102, 9 ], [ 5, 102, 17 ],
      [ 5, 105, 5 ], [ 5, 109, 5 ], [ 5, 109, 14 ], [ 5, 112, 5 ], [ 5, 112, 7 ], [ 5, 113, 4 ],
      [ 5, 116, 20 ], [ 5, 118, 18 ], [ 5, 119, 4 ], [ 5, 119, 5 ], [ 5, 119, 10 ],
      [ 5, 119, 12 ], [ 5, 120, 5 ], [ 5, 120, 14 ], [ 5, 121, 13 ], [ 5, 122, 9 ],
      [ 5, 122, 15 ], [ 5, 127, 11 ], [ 5, 134, 9 ], [ 5, 136, 18 ], [ 5, 140, 23 ],
      [ 5, 142, 14 ], [ 5, 143, 23 ], [ 5, 148, 5 ], [ 5, 154, 15 ], [ 5, 160, 4 ],
      [ 5, 160, 7 ], [ 5, 161, 5 ], [ 5, 161, 7 ], [ 5, 162, 5 ], [ 5, 224, 9 ], [ 5, 227, 11 ],
      [ 5, 232, 14 ], [ 5, 242, 9 ], [ 5, 526, 11 ], [ 5, 534, 11 ], [ 5, 536, 13 ],
      [ 5, 546, 11 ], [ 5, 580, 2 ], [ 5, 604, 2 ], [ 5, 604, 4 ], [ 5, 605, 2 ], [ 5, 665, 4 ],
      [ 5, 666, 4 ], [ 5, 668, 2 ], [ 5, 670, 3 ], [ 5, 671, 2 ], [ 5, 672, 2 ], [ 5, 673, 2 ],
      [ 5, 704, 3 ], [ 5, 706, 8 ], [ 5, 708, 2 ], [ 5, 709, 3 ], [ 5, 713, 2 ], [ 5, 714, 2 ],
      [ 5, 715, 2 ], [ 5, 716, 2 ], [ 5, 717, 2 ], [ 5, 719, 2 ], [ 5, 720, 2 ], [ 5, 721, 2 ],
      [ 5, 763, 3 ], [ 5, 770, 2 ], [ 5, 774, 4 ], [ 5, 948, 1 ], [ 5, 948, 2 ], [ 5, 948, 3 ],
      [ 5, 948, 4 ], [ 5, 952, 1 ], [ 5, 952, 3 ] ]
    gap> List(N5H1FC2,x->StructureDescription(N5g[x]));
    [ "C2 x C2", "C2 x C2", "C2 x C2 x C2", "C2 x C2 x C2", "C2 x C2 x C2", "C2 x C2 x C2",
      "C2 x C2 x C2 x C2", "C2 x C2 x C2 x C2", "C2 x C2 x C2 x C2", "C2 x C2 x C2 x C2",
      "C2 x C2 x C2 x C2", "C2 x C2 x C2 x C2", "C2 x C2 x C2 x C2", "C2 x C2 x C2",
      "C2 x C2 x C2", "C2 x C2 x C2", "C2 x C2 x C2", "C2 x C2 x C2", "C2 x C2 x C2",
      "C2 x C2 x C2", "C2 x C2 x C2", "C2 x C2 x C2", "C2 x C2 x C2", "C2 x C2 x C2", "D8",
      "C2 x D8", "C2 x D8", "C2 x D8", "C2 x D8", "C2 x D8", "C2 x D8", "C2 x D8", "C2 x D8",
      "C2 x D8", "C2 x D8", "C2 x D8", "C2 x D8", "C2 x D8", "C2 x D8", "C2 x D8", "C2 x D8",
      "C2 x D8", "C2 x D8", "C2 x D8", "C2 x D8", "C2 x D8", "C2 x D8", "C2 x D8", "C2 x D8",
      "C2 x D8", "C2 x D8", "C4 x C2 x C2", "C4 x C2 x C2", "C4 x C2 x C2", "C4 x C2 x C2",
      "C4 x C2 x C2", "C2 x C2 x D8", "C2 x C2 x D8", "C2 x C2 x D8", "C2 x C2 x D8",
      "C2 x C2 x D8", "D8", "D8", "D8", "D8", "D8", "D8", "C4 x C2", "C4 x C2", "C4 x C2",
      "C4 x C2", "C4 : C4", "(C4 x C2) : C2", "(C4 x C2) : C2", "C4 x C2 x C2", "C4 x C2 x C2",
      "C4 x C4", "(C4 x C2) : C2", "(C4 x C2) : C2", "C2 x D8", "C2 x D8", "C2 x D8", "C2 x D8",
      "C2 x D8", "C2 x D8", "C2 x D8", "C2 x D8", "C2 x D8", "C4 x D8", "(C4 x C4) : C2",
      "(C2 x C2 x C2 x C2) : C2", "(C4 x C2 x C2) : C2", "(C2 x C2 x C2 x C2) : C2",
      "(C4 x C2 x C2) : C2", "C2 x C2 x D8", "D8 x D8", "C4 x C2", "C4 x C2", "C4 x C2",
      "C4 x C2", "C4 x C2", "C6 x C2", "D12", "D12", "C2 x C2 x S3", "C2 x C2 x A4", "C2 x S4",
      "C2 x S4", "C2 x C2 x S4", "A4", "C2 x A4", "C2 x A4", "C2 x A4", "C8 : C2", "C8 : C2",
      "(C4 x C2) : C2", "(C4 x C2) : C2", "(C4 x C2) : C2", "QD16", "QD16", "(C2 x C2 x C2) : C4",
      "(C2 x C2 x C2) : C4", "(C2 x C2 x C2) : (C2 x C2)", "(C2 x C2 x C2) : (C2 x C2)",
      "C8 : (C2 x C2)", "C8 : (C2 x C2)", "C8 : (C2 x C2)", "C8 : (C2 x C2)", "C8 : (C2 x C2)",
      "C2 x ((C4 x C2) : C2)", "C2 x (C8 : C2)", "C2 x QD16", "((C2 x C2 x C2) : (C2 x C2)) : C2",
      "C2 x (C8 : (C2 x C2))", "Q8", "C2 x A5", "C2 x A5", "C2 x A5", "C2 x A5", "A5", "A5" ]

    # N41 is the list of CARAT IDs of decomposable lattice of rank 5=4+1
    # whose flabby class [M_G]^fl is not invertible [HY17, Example 4.12]
    gap> N41g:=List(N41,x->CaratMatGroupZClass(x[1],x[2],x[3]));;
    gap> Length(N41g); # there exist 768 not retract rational tori in dim=5=4+1 [HY17, Table 14]
    768
    gap> N41gF:=List(N41g,x->FlabbyResolutionLowRank(x).actionF);;
    gap> N41H1F=List(N41gF,x->Filtered(H1(x),y->y>1));;
    gap> Collected(N41H1F);
    [ [ [  ], 690 ], [ [ 2 ], 73 ], [ [ 2, 2 ], 5 ] ]

    gap> N41H1FC2xC2:=Filtered([1..Length(N41H1F)],x->N41H1F[x]=[2,2]);
    [ 589, 590, 591, 720, 721 ]
    gap> List(N41H1FC2xC2,x->N41[x]);
    [ [ 5, 664, 1 ], [ 5, 773, 3 ], [ 5, 774, 3 ], [ 5, 691, 1 ], [ 5, 730, 1 ] ]
    gap> List(N41H1FC2xC2,x->StructureDescription(N41g[x]));
    [ "C2 x Q8", "Q8", "Q8", "SL(2,3)", "C2 x SL(2,3)" ]

    gap> N41H1FC2:=Filtered([1..Length(N41H1F)],x->N41H1F[x]=[2]);
    [ 1, 2, 3, 4, 9, 10, 11, 12, 13, 14, 36, 37, 38, 39, 40, 41, 42, 74, 75, 76, 77, 93, 94, 95,
      96, 97, 112, 113, 114, 115, 116, 117, 118, 254, 255, 256, 257, 281, 282, 283, 284, 285,
      592, 593, 594, 595, 596, 597, 598, 599, 600, 601, 602, 603, 604, 605, 610, 611, 612, 613,
      614, 615, 616, 727, 728, 729, 730, 731, 732, 738, 739, 740, 741 ]
    gap> List(N41H1FC2,x->N41[x]); # CARAT ID’s of F with H1(F)=C2
    [ [ 5, 18, 18 ], [ 5, 18, 21 ], [ 5, 19, 10 ], [ 5, 32, 23 ], [ 5, 20, 10 ], [ 5, 20, 13 ],
      [ 5, 25, 14 ], [ 5, 30, 14 ], [ 5, 31, 16 ], [ 5, 31, 29 ], [ 5, 21, 10 ], [ 5, 24, 18 ],
      [ 5, 24, 21 ], [ 5, 26, 19 ], [ 5, 31, 22 ], [ 5, 31, 25 ], [ 5, 32, 30 ], [ 5, 66, 5 ],
      [ 5, 83, 7 ], [ 5, 101, 4 ], [ 5, 102, 4 ], [ 5, 63, 12 ], [ 5, 65, 12 ], [ 5, 76, 31 ],
      [ 5, 99, 5 ], [ 5, 100, 5 ], [ 5, 48, 12 ], [ 5, 71, 8 ], [ 5, 72, 26 ], [ 5, 75, 26 ],
      [ 5, 78, 26 ], [ 5, 79, 26 ], [ 5, 88, 26 ], [ 5, 112, 3 ], [ 5, 160, 3 ], [ 5, 161, 3 ],
      [ 5, 162, 3 ], [ 5, 119, 3 ], [ 5, 120, 11 ], [ 5, 121, 11 ], [ 5, 122, 14 ],
      [ 5, 148, 3 ], [ 5, 665, 3 ], [ 5, 666, 3 ], [ 5, 667, 3 ], [ 5, 720, 1 ], [ 5, 672, 1 ],
      [ 5, 673, 1 ], [ 5, 674, 1 ], [ 5, 675, 1 ], [ 5, 721, 1 ], [ 5, 668, 1 ], [ 5, 669, 1 ],
      [ 5, 670, 1 ], [ 5, 671, 1 ], [ 5, 719, 1 ], [ 5, 713, 1 ], [ 5, 714, 1 ], [ 5, 715, 1 ],
      [ 5, 716, 1 ], [ 5, 717, 1 ], [ 5, 718, 1 ], [ 5, 770, 1 ], [ 5, 731, 1 ], [ 5, 732, 1 ],
      [ 5, 775, 1 ], [ 5, 733, 1 ], [ 5, 734, 1 ], [ 5, 776, 1 ], [ 5, 682, 1 ], [ 5, 780, 1 ],
      [ 5, 781, 1 ], [ 5, 783, 1 ] ]
    gap> List(N41H1FC2,x->StructureDescription(N41g[x]));
    [ "C2 x C2", "C2 x C2", "C2 x C2", "C2 x C2 x C2", "C2 x C2 x C2", "C2 x C2 x C2",
      "C2 x C2 x C2 x C2", "C2 x C2 x C2", "C2 x C2 x C2", "C2 x C2 x C2", "C2 x C2 x C2",
      "C2 x C2 x C2", "C2 x C2 x C2", "C2 x C2 x C2 x C2", "C2 x C2 x C2", "C2 x C2 x C2",
      "C2 x C2 x C2", "C4 x C2", "C4 x C2 x C2", "C4 x C2", "C4 x C2", "D8", "D8", "C2 x D8",
      "D8", "D8", "C2 x D8", "C2 x D8", "C2 x D8", "C2 x D8", "C2 x D8", "C2 x D8",
      "C2 x C2 x D8", "C4 x C2 x C2", "C4 x C2", "C4 x C2", "C4 x C2", "C2 x D8", "C2 x D8",
      "C2 x D8", "C2 x D8", "C2 x C2 x D8", "C8 : C2", "C8 : C2", "C8 : C2", "C2 x (C8 : C2)",
      "QD16", "QD16", "QD16", "QD16", "C2 x QD16", "(C4 x C2) : C2", "(C4 x C2) : C2",
      "(C4 x C2) : C2", "(C4 x C2) : C2", "C2 x ((C4 x C2) : C2)", "C8 : (C2 x C2)",
      "C8 : (C2 x C2)", "C8 : (C2 x C2)", "C8 : (C2 x C2)", "C8 : (C2 x C2)", "C8 : (C2 x C2)",
      "C2 x (C8 : (C2 x C2))", "((C4 x C2) : C2) : C3", "((C4 x C2) : C2) : C3",
      "C2 x (((C4 x C2) : C2) : C3)", "GL(2,3)", "GL(2,3)", "C2 x GL(2,3)", "C2 x (GL(2,3) : C2)",
      "(((C4 x C2) : C2) : C3) : C2", "(((C4 x C2) : C2) : C3) : C2",
      "(((C4 x C2) : C2) : C3) : C2" ]

    # N32 is the list of CARAT IDs of decomposable lattice of rank 5=3+2
    # whose flabby class [M_G]^fl is not invertible [HY17, Example 4.12]
    gap> N32g:=List(N32,x->CaratMatGroupZClass(x[1],x[2],x[3]));;
    gap> Length(N32g); # there exist 849 not retract rational tori in dim=5=3+2 [HY17, Table 13]
    849
    gap> N32gF:=List(N32g,x->FlabbyResolutionLowRank(x).actionF);;
    gap> N32H1F:=List(N32gF,x->Filtered(H1(x),y->y>1));;
    gap> Collected(N32H1F);
    [ [ [  ], 813 ], [ [ 2 ], 36 ] ]]

    gap> N32H1FC2:=Filtered([1..Length(N32H1F)],x->N32H1F[x]=[2]);
    [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
      22, 23, 24, 25, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588 ]
    gap> List(N32H1FC2,x->N32[x]); # CARAT ID’s of F with H1(F)=C2
    [ [ 5, 14, 8 ], [ 5, 18, 19 ], [ 5, 24, 19 ], [ 5, 26, 20 ], [ 5, 31, 17 ],
      [ 5, 32, 17 ], [ 5, 78, 8 ], [ 5, 78, 27 ], [ 5, 80, 8 ], [ 5, 86, 5 ],
      [ 5, 93, 5 ], [ 5, 100, 11 ], [ 5, 102, 8 ], [ 5, 224, 4 ], [ 5, 227, 5 ],
      [ 5, 228, 3 ], [ 5, 232, 4 ], [ 5, 232, 9 ], [ 5, 237, 3 ], [ 5, 242, 4 ],
      [ 5, 242, 14 ], [ 5, 247, 3 ], [ 5, 247, 7 ], [ 5, 253, 4 ],
      [ 5, 259, 3 ], [ 5, 520, 17 ], [ 5, 525, 2 ], [ 5, 560, 3 ],
      [ 5, 566, 3 ], [ 5, 580, 1 ], [ 5, 590, 1 ], [ 5, 605, 1 ], [ 5, 620, 1 ],
      [ 5, 629, 1 ], [ 5, 634, 1 ], [ 5, 634, 3 ] ]
    gap> List(N32H1FC2,x->StructureDescription(N32g[x]));
    [ "C2 x C2", "C2 x C2", "C2 x C2 x C2", "C2 x C2 x C2 x C2", "C2 x C2 x C2",
      "C2 x C2 x C2", "C2 x D8", "C2 x D8", "C2 x D8", "C4 x C2 x C2",
      "C2 x C2 x D8", "D8", "C4 x C2", "C6 x C2", "D12", "C6 x C2", "D12",
      "D12", "C6 x C2 x C2", "C2 x C2 x S3", "C2 x C2 x S3", "C2 x C2 x S3",
      "C2 x C2 x S3", "C2 x C2 x S3", "C2 x C2 x C2 x S3", "C2 x A4",
      "C2 x C2 x A4", "C4 x A4", "A4 x D8", "A4", "C2 x A4 x S3", "C2 x A4",
      "C3 x A4", "C6 x A4", "A4 x S3", "A4 x S3" ]

    # N311 is the list of CARAT IDs of decomposable lattice of rank 5=3+1+1
    # whose flabby class [M_G]^fl is not invertible [HY17, Example 4.12]
    gap> N311g:=List(N311,x->CaratMatGroupZClass(x[1],x[2],x[3]));;
    gap> Length(N311g); # there exist 245 not retract rational tori in dim=5=3+1+1 [HY17, Table 12]
    245
    gap> N311gF:=List(N311g,x->FlabbyResolutionLowRank(x).actionF);;
    gap> N311H1F:=List(N311gF,x->Filtered(H1(x),y->y>1));;
    gap> Collected(N311H1F);
    [ [ [  ], 232 ], [ [ 2 ], 13 ] ]

    gap> N311H1FC2:=Filtered([1..Length(N311H1F)],x->N311H1F[x]=[2]);
    [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 164, 165, 166, 167 ]
    gap> List(N311H1FC2,x->N311[x]); # CARAT ID’s of F with H1(F)=C2
    [ [ 5, 11, 4 ], [ 5, 14, 4 ], [ 5, 18, 7 ], [ 5, 19, 5 ], [ 5, 21, 5 ],
      [ 5, 24, 7 ], [ 5, 26, 3 ], [ 5, 31, 4 ], [ 5, 32, 10 ], [ 5, 502, 6 ],
      [ 5, 505, 1 ], [ 5, 520, 16 ], [ 5, 525, 1 ] ]
    gap> List(N311H1FC2,x->StructureDescription(N311g[x]));
    [ "C2 x C2", "C2 x C2", "C2 x C2", "C2 x C2", "C2 x C2 x C2",
      "C2 x C2 x C2", "C2 x C2 x C2 x C2", "C2 x C2 x C2", "C2 x C2 x C2",
      "A4", "C2 x A4", "C2 x A4", "C2 x C2 x A4" ]

##  5\. Proof of Theorem 1.15 and Theorem 1.17

Let $T=R^{(1)}_{K/k}(\mathbbm{G}_{m})$ be a norm one torus of $K/k$. We have
the character module $\widehat{T}=J_{G/H}$ of $T$ and then $H^{1}(k,{\rm
Pic}\,\overline{X})\simeq H^{1}(G,[J_{G/H}]^{fl})$ (see Section 2). We may
assume that $H$ is the stabilizer of one of the letters in $G$, i.e.
$L=k(\theta_{1},\ldots,\theta_{n})$ and $K=k(\theta_{i})$ for some $1\leq
i\leq n$. In order to compute $H^{1}(G,[J_{G/H}]^{fl})$, we apply the
functions Norm1TorusJ($n,m$) and
FlabbyResolutionLowRankFromGroup($G$,$nTm$).actionF in [HHY20, Algorithm 4.1].
Norm1TorusJ($n,m$) returns $J_{G/H}$ for $G=nTm\leq S_{n}$ and $H$ is the
stabilizer of one of the letters in $G$ and
FlabbyResolutionLowRankFromGroup($G$,$nTm$).actionF returns a suitable flabby
class $F=[J_{G/H}]^{fl}$ with low rank by using the backtracking techniques
for $G=nTm\leq S_{n}$.

Proof of Theorem 1.15.

For $G=nTm$ $(2\leq n\leq 11)$, the computation is described in Example 5.1.

For $G=nTm$ $(12\leq n\leq 15)$, it needs much time and computer resources
(memory) in computations. At present, we do not know the complete solutions
for $n=12$.

For $13\leq n\leq 15$, we wish to compute $H^{1}(G,[J_{G/H}]^{fl})$ for each
cases $G=13Tm$ $(1\leq m\leq 9)$, $G=14Tm$ $(1\leq m\leq 63)$, $G=15Tm$
$(1\leq m\leq 104)$. This is achievable except for the each last two groups
$G=13T8\simeq A_{13}$, $13T9\simeq S_{13}$, $14T62\simeq A_{14}$, $14T63\simeq
S_{14}$, $15T103\simeq A_{15}$, $15T104\simeq S_{15}$ because of the computer
resources reason (see Example 5.2). However, for the exceptional cases, we
already know that $H^{1}(G,[J_{G/H}]^{fl})=0$ by Theorem 1.10 and Theorem
1.11.

The last assertion follows from Theorem 1.2.∎

Proof of Theorem 1.17.

For $G=11T6\simeq M_{11}\leq S_{11}$ and $G=23T5\simeq M_{23}\leq S_{23}$, it
follows from Theorem 2.1 that $H^{1}(G,[J_{G/H}]^{fl})=0$ (see also the
paragraph after Theorem 3.6). For $G=24T24680\simeq M_{24}\leq S_{24}$, we
know that the Schur multiplier of $G$ vanishes: $M(G)\simeq
H^{3}(G,\mathbbm{Z})=0$ (see Mazet [Maz82]). We know that the Mathieu groups
are simple groups and a subgroup $H\leq M_{24}$ with $[M_{24}:H]=24$ is
isomorphic to $M_{23}$ which is the stabilizer of one of the letters in
$M_{24}$ (see e.g. [DM96, Exercises 6.8.8]). Hence it follows from
$0=H^{ab}\simeq H^{2}(H,\mathbbm{Z})\simeq H^{2}(G,\mathbbm{Z}[G/H])\to
H^{2}(G,J_{G/H})\xrightarrow{\delta}H^{3}(G,\mathbbm{Z})=0$ that
$H^{2}(G,J_{G/H})=0$. Thus we have
$H^{1}(G,[J_{G/H}]^{fl})\simeq{}^{2}_{\omega}(G,J_{G/H})=0$.

For $G=12T295\simeq M_{12}\leq S_{12}$ and $G=22T38\simeq M_{22}\leq S_{22}$,
the computation is described in Example 5.3. ∎

Some related functions for Example 5.1, Example 5.2 and Example 5.3 are
available from
https://www.math.kyoto-u.ac.jp/~yamasaki/Algorithm/RatProbNorm1Tori/.

######  Example 5.1 (Computation of $H^{1}(G,[J_{G/H}]^{fl})$ where $G=nTm$
$(n\leq 11)$).

    gap> Read("FlabbyResolutionFromBase.gap");
    gap> for n in [2..11] do for m in [1..NrTransitiveGroups(n)] do
    > F:=FlabbyResolutionLowRankFromGroup(Norm1TorusJ(n,m),TransitiveGroup(n,m)).actionF;
    > Print([[n,m],Length(F.1),Filtered(H1(F),x->x>1)],"\n");od;Print("\n");od;
    [ [ 2, 1 ], 1, [  ] ]

    [ [ 3, 1 ], 1, [  ] ]
    [ [ 3, 2 ], 4, [  ] ]

    [ [ 4, 1 ], 1, [  ] ]
    [ [ 4, 2 ], 5, [ 2 ] ]
    [ [ 4, 3 ], 7, [  ] ]
    [ [ 4, 4 ], 9, [ 2 ] ]
    [ [ 4, 5 ], 15, [  ] ]

    [ [ 5, 1 ], 1, [  ] ]
    [ [ 5, 2 ], 6, [  ] ]
    [ [ 5, 3 ], 16, [  ] ]
    [ [ 5, 4 ], 16, [  ] ]
    [ [ 5, 5 ], 16, [  ] ]

    [ [ 6, 1 ], 1, [  ] ]
    [ [ 6, 2 ], 7, [  ] ]
    [ [ 6, 3 ], 9, [  ] ]
    [ [ 6, 4 ], 10, [ 2 ] ]
    [ [ 6, 5 ], 21, [  ] ]
    [ [ 6, 6 ], 10, [  ] ]
    [ [ 6, 7 ], 19, [  ] ]
    [ [ 6, 8 ], 19, [  ] ]
    [ [ 6, 9 ], 27, [  ] ]
    [ [ 6, 10 ], 27, [  ] ]
    [ [ 6, 11 ], 19, [  ] ]
    [ [ 6, 12 ], 10, [ 2 ] ]
    [ [ 6, 13 ], 27, [  ] ]
    [ [ 6, 14 ], 31, [  ] ]
    [ [ 6, 15 ], 60, [  ] ]
    [ [ 6, 16 ], 60, [  ] ]

    [ [ 7, 1 ], 1, [  ] ]
    [ [ 7, 2 ], 8, [  ] ]
    [ [ 7, 3 ], 15, [  ] ]
    [ [ 7, 4 ], 36, [  ] ]
    [ [ 7, 5 ], 15, [  ] ]
    [ [ 7, 6 ], 36, [  ] ]
    [ [ 7, 7 ], 36, [  ] ]

    [ [ 8, 1 ], 1, [  ] ]
    [ [ 8, 2 ], 9, [ 2 ] ]
    [ [ 8, 3 ], 17, [ 2, 2, 2 ] ]
    [ [ 8, 4 ], 9, [ 2 ] ]
    [ [ 8, 5 ], 9, [  ] ]
    [ [ 8, 6 ], 11, [  ] ]
    [ [ 8, 7 ], 21, [  ] ]
    [ [ 8, 8 ], 11, [  ] ]
    [ [ 8, 9 ], 21, [ 2 ] ]
    [ [ 8, 10 ], 21, [  ] ]
    [ [ 8, 11 ], 21, [ 2 ] ]
    [ [ 8, 12 ], 25, [  ] ]
    [ [ 8, 13 ], 19, [ 2 ] ]
    [ [ 8, 14 ], 13, [ 2 ] ]
    [ [ 8, 15 ], 43, [ 2 ] ]
    [ [ 8, 16 ], 29, [  ] ]
    [ [ 8, 17 ], 43, [  ] ]
    [ [ 8, 18 ], 91, [  ] ]
    [ [ 8, 19 ], 51, [ 2 ] ]
    [ [ 8, 20 ], 29, [  ] ]
    [ [ 8, 21 ], 49, [ 2 ] ]
    [ [ 8, 22 ], 49, [ 2 ] ]
    [ [ 8, 23 ], 31, [  ] ]
    [ [ 8, 24 ], 31, [  ] ]
    [ [ 8, 25 ], 49, [  ] ]
    [ [ 8, 26 ], 67, [  ] ]
    [ [ 8, 27 ], 29, [  ] ]
    [ [ 8, 28 ], 83, [  ] ]
    [ [ 8, 29 ], 99, [  ] ]
    [ [ 8, 30 ], 67, [  ] ]
    [ [ 8, 31 ], 49, [ 2 ] ]
    [ [ 8, 32 ], 61, [ 2 ] ]
    [ [ 8, 33 ], 99, [  ] ]
    [ [ 8, 34 ], 123,[  ] ]
    [ [ 8, 35 ], 99, [  ] ]
    [ [ 8, 36 ], 49, [  ] ]
    [ [ 8, 37 ], 49, [ 2 ] ]
    [ [ 8, 38 ], 61, [ 2 ] ]
    [ [ 8, 39 ], 211, [  ] ]
    [ [ 8, 40 ], 115, [  ] ]
    [ [ 8, 41 ], 123, [  ] ]
    [ [ 8, 42 ], 123, [  ] ]
    [ [ 8, 43 ], 91, [  ] ]
    [ [ 8, 44 ], 211, [  ] ]
    [ [ 8, 45 ], 123, [  ] ]
    [ [ 8, 46 ], 123, [  ] ]
    [ [ 8, 47 ], 123, [  ] ]
    [ [ 8, 48 ], 483, [  ] ]
    [ [ 8, 49 ], 539, [  ] ]
    [ [ 8, 50 ], 539, [  ] ]

    [ [ 9, 1 ], 1, [  ] ]
    [ [ 9, 2 ], 10, [ 3 ] ]
    [ [ 9, 3 ], 10, [  ] ]
    [ [ 9, 4 ], 13, [  ] ]
    [ [ 9, 5 ], 28, [ 3 ] ]
    [ [ 9, 6 ], 31, [  ] ]
    [ [ 9, 7 ], 31, [ 3 ] ]
    [ [ 9, 8 ], 28, [  ] ]
    [ [ 9, 9 ], 28, [ 3 ] ]
    [ [ 9, 10 ], 70, [  ] ]
    [ [ 9, 11 ], 70, [ 3 ] ]
    [ [ 9, 12 ], 61, [  ] ]
    [ [ 9, 13 ], 40, [  ] ]
    [ [ 9, 14 ], 64, [ 3 ] ]
    [ [ 9, 15 ], 64, [  ] ]
    [ [ 9, 16 ], 34, [  ] ]
    [ [ 9, 17 ], 31, [  ] ]
    [ [ 9, 18 ], 70, [  ] ]
    [ [ 9, 19 ], 64, [  ] ]
    [ [ 9, 20 ], 61, [  ] ]
    [ [ 9, 21 ], 70, [  ] ]
    [ [ 9, 22 ], 40, [  ] ]
    [ [ 9, 23 ], 88, [ 3 ] ]
    [ [ 9, 24 ], 70, [  ] ]
    [ [ 9, 25 ], 40, [  ] ]
    [ [ 9, 26 ], 88, [  ] ]
    [ [ 9, 27 ], 64, [  ] ]
    [ [ 9, 28 ], 40, [  ] ]
    [ [ 9, 29 ], 70, [  ] ]
    [ [ 9, 30 ], 70, [  ] ]
    [ [ 9, 31 ], 70, [  ] ]
    [ [ 9, 32 ], 232, [  ] ]
    [ [ 9, 33 ], 1744, [  ] ]
    [ [ 9, 34 ], 1744, [  ] ]

    [ [ 10, 1 ], 1, [  ] ]
    [ [ 10, 2 ], 11, [  ] ]
    [ [ 10, 3 ], 13, [  ] ]
    [ [ 10, 4 ], 13, [  ] ]
    [ [ 10, 5 ], 31, [  ] ]
    [ [ 10, 6 ], 53, [  ] ]
    [ [ 10, 7 ], 26, [ 2 ] ]
    [ [ 10, 8 ], 36, [  ] ]
    [ [ 10, 9 ], 63, [  ] ]
    [ [ 10, 10 ], 63, [  ] ]
    [ [ 10, 11 ], 31, [  ] ]
    [ [ 10, 12 ], 31, [  ] ]
    [ [ 10, 13 ], 36, [  ] ]
    [ [ 10, 14 ], 36, [  ] ]
    [ [ 10, 15 ], 51, [  ] ]
    [ [ 10, 16 ], 51, [  ] ]
    [ [ 10, 17 ], 83, [  ] ]
    [ [ 10, 18 ], 83, [  ] ]
    [ [ 10, 19 ], 83, [  ] ]
    [ [ 10, 20 ], 83, [  ] ]
    [ [ 10, 21 ], 63, [  ] ]
    [ [ 10, 22 ], 31, [  ] ]
    [ [ 10, 23 ], 51, [  ] ]
    [ [ 10, 24 ], 61, [  ] ]
    [ [ 10, 25 ], 61, [  ] ]
    [ [ 10, 26 ], 46, [ 2 ] ]
    [ [ 10, 27 ], 83, [  ] ]
    [ [ 10, 28 ], 83, [  ] ]
    [ [ 10, 29 ], 61, [  ] ]
    [ [ 10, 30 ], 91, [  ] ]
    [ [ 10, 31 ], 67, [  ] ]
    [ [ 10, 32 ], 46, [ 2 ] ]
    [ [ 10, 33 ], 83, [  ] ]
    [ [ 10, 34 ], 61, [  ] ]
    [ [ 10, 35 ], 91, [  ] ]
    [ [ 10, 36 ], 61, [  ] ]
    [ [ 10, 37 ], 61, [  ] ]
    [ [ 10, 38 ], 61, [  ] ]
    [ [ 10, 39 ], 61, [  ] ]
    [ [ 10, 40 ], 83, [  ] ]
    [ [ 10, 41 ], 83, [  ] ]
    [ [ 10, 42 ], 83, [  ] ]
    [ [ 10, 43 ], 83, [  ] ]
    [ [ 10, 44 ], 378, [  ] ]
    [ [ 10, 45 ], 378, [  ] ]

    [ [ 11, 1 ], 1, [  ] ]
    [ [ 11, 2 ], 12, [  ] ]
    [ [ 11, 3 ], 45, [  ] ]
    [ [ 11, 4 ], 100, [  ] ]
    [ [ 11, 5 ], 56, [  ] ]
    [ [ 11, 6 ], 100, [  ] ]
    [ [ 11, 7 ], 100, [  ] ]
    [ [ 11, 8 ], 100, [  ] ]

######  Example 5.2 (Computation of $H^{1}(G,[J_{G/H}]^{fl})$ where $G=13Tm$
$(1\leq m\leq 9,m\neq 8,9)$, $G=14Tm$ $(1\leq m\leq 63,m\neq 62,63)$, $G=15Tm$
$(1\leq m\leq 104,m\neq 103,104)$).

    gap> Read("FlabbyResolutionFromBase.gap");
    gap> for m in [1..NrTransitiveGroups(13)-2] do
    > F:=FlabbyResolutionLowRankFromGroup(Norm1TorusJ(13,m),TransitiveGroup(13,m)).actionF;
    > Print([[n,i],Length(F.1),Filtered(H1(F),x->x>1)],"\n");od;Print("\n");od;
    [ [ 13, 1 ], 1, [  ] ]
    [ [ 13, 2 ], 14, [  ] ]
    [ [ 13, 3 ], 27, [  ] ]
    [ [ 13, 4 ], 40, [  ] ]
    [ [ 13, 5 ], 66, [  ] ]
    [ [ 13, 6 ], 144, [  ] ]
    [ [ 13, 7 ], 40, [  ] ]

    gap> for m in [1..NrTransitiveGroups(14)-2] do
    > F:=FlabbyResolutionLowRankFromGroup(Norm1TorusJ(14,m),TransitiveGroup(14,m)).actionF;
    > Print([[14,m],Length(F.1),Filtered(H1(F),x->x>1)],"\n");od;
    [ [ 14, 1 ], 1, [  ] ]
    [ [ 14, 2 ], 15, [  ] ]
    [ [ 14, 3 ], 17, [  ] ]
    [ [ 14, 4 ], 31, [  ] ]
    [ [ 14, 5 ], 31, [  ] ]
    [ [ 14, 6 ], 50, [  ] ]
    [ [ 14, 7 ], 57, [  ] ]
    [ [ 14, 8 ], 101, [  ] ]
    [ [ 14, 9 ], 78, [  ] ]
    [ [ 14, 10 ], 64, [  ] ]
    [ [ 14, 11 ], 86, [  ] ]
    [ [ 14, 12 ], 115, [  ] ]
    [ [ 14, 13 ], 115, [  ] ]
    [ [ 14, 14 ], 129, [  ] ]
    [ [ 14, 15 ], 129, [  ] ]
    [ [ 14, 16 ], 31, [  ] ]
    [ [ 14, 17 ], 92, [  ] ]
    [ [ 14, 18 ], 92, [  ] ]
    [ [ 14, 19 ], 31, [  ] ]
    [ [ 14, 20 ], 115, [  ] ]
    [ [ 14, 21 ], 78, [  ] ]
    [ [ 14, 22 ], 171, [  ] ]
    [ [ 14, 23 ], 171, [  ] ]
    [ [ 14, 24 ], 171, [  ] ]
    [ [ 14, 25 ], 171, [  ] ]
    [ [ 14, 26 ], 129, [  ] ]
    [ [ 14, 27 ], 99, [  ] ]
    [ [ 14, 28 ], 99, [  ] ]
    [ [ 14, 29 ], 78, [  ] ]
    [ [ 14, 30 ], 92, [ 2 ] ]
    [ [ 14, 31 ], 171, [  ] ]
    [ [ 14, 32 ], 171, [  ] ]
    [ [ 14, 33 ], 92, [  ] ]
    [ [ 14, 34 ], 92, [  ] ]
    [ [ 14, 35 ], 92, [  ] ]
    [ [ 14, 36 ], 171, [  ] ]
    [ [ 14, 37 ], 171, [  ] ]
    [ [ 14, 38 ], 99, [  ] ]
    [ [ 14, 39 ], 183, [  ] ]
    [ [ 14, 40 ], 127, [  ] ]
    [ [ 14, 41 ], 127,[  ] ]
    [ [ 14, 42 ], 92, [  ] ]
    [ [ 14, 43 ], 92, [  ] ]
    [ [ 14, 44 ], 99, [  ] ]
    [ [ 14, 45 ], 171, [  ] ]
    [ [ 14, 46 ], 57, [  ] ]
    [ [ 14, 47 ], 57, [  ] ]
    [ [ 14, 48 ], 127, [  ] ]
    [ [ 14, 49 ], 57, [  ] ]
    [ [ 14, 50 ], 92, [  ] ]
    [ [ 14, 51 ], 92, [  ] ]
    [ [ 14, 52 ], 129, [  ] ]
    [ [ 14, 53 ], 127, [  ] ]
    [ [ 14, 54 ], 127, [  ] ]
    [ [ 14, 55 ], 127, [  ] ]
    [ [ 14, 56 ], 127, [  ] ]
    [ [ 14, 57 ], 127, [  ] ]
    [ [ 14, 58 ], 171, [  ] ]
    [ [ 14, 59 ], 171, [  ] ]
    [ [ 14, 60 ], 171, [  ] ]
    [ [ 14, 61 ], 171, [  ] ]

    gap> for m in [1..NrTransitiveGroups(15)-2] do
    > F:=FlabbyResolutionLowRankFromGroup(Norm1TorusJ(15,m),TransitiveGroup(15,m)).actionF;
    > Print([[15,m],Length(F.1),Filtered(H1(F),x->x>1)],"\n");od;
    [ [ 15, 1 ], 1, [  ] ]
    [ [ 15, 2 ], 16, [  ] ]
    [ [ 15, 3 ], 14, [  ] ]
    [ [ 15, 4 ], 21, [  ] ]
    [ [ 15, 5 ], 21, [  ] ]
    [ [ 15, 6 ], 39, [  ] ]
    [ [ 15, 7 ], 17, [  ] ]
    [ [ 15, 8 ], 36, [  ] ]
    [ [ 15, 9 ], 79, [ 5 ] ]
    [ [ 15, 10 ], 36, [  ] ]
    [ [ 15, 11 ], 27, [  ] ]
    [ [ 15, 12 ], 94, [  ] ]
    [ [ 15, 13 ], 82, [  ] ]
    [ [ 15, 14 ], 97, [ 5 ] ]
    [ [ 15, 15 ], 51, [  ] ]
    [ [ 15, 16 ], 36, [  ] ]
    [ [ 15, 17 ], 127, [  ] ]
    [ [ 15, 18 ], 97, [  ] ]
    [ [ 15, 19 ], 124, [  ] ]
    [ [ 15, 20 ], 81, [  ] ]
    [ [ 15, 21 ], 66, [  ] ]
    [ [ 15, 22 ], 39, [  ] ]
    [ [ 15, 23 ], 27, [  ] ]
    [ [ 15, 24 ], 36, [  ] ]
    [ [ 15, 25 ], 79, [  ] ]
    [ [ 15, 26 ], 96, [  ] ]
    [ [ 15, 27 ], 127, [  ] ]
    [ [ 15, 28 ], 81, [  ] ]
    [ [ 15, 29 ], 27, [  ] ]
    [ [ 15, 30 ], 94, [  ] ]
    [ [ 15, 31 ], 169, [  ] ]
    [ [ 15, 32 ], 154, [  ] ]
    [ [ 15, 33 ], 111, [  ] ]
    [ [ 15, 34 ], 186, [  ] ]
    [ [ 15, 35 ], 201, [  ] ]
    [ [ 15, 36 ], 96, [  ] ]
    [ [ 15, 37 ], 199, [  ] ]
    [ [ 15, 38 ], 124, [  ] ]
    [ [ 15, 39 ], 94, [  ] ]
    [ [ 15, 40 ], 169, [  ] ]
    [ [ 15, 41 ], 186, [  ] ]
    [ [ 15, 42 ], 201, [  ] ]
    [ [ 15, 43 ], 201, [  ] ]
    [ [ 15, 44 ], 111, [  ] ]
    [ [ 15, 45 ], 201, [  ] ]
    [ [ 15, 46 ], 186, [  ] ]
    [ [ 15, 47 ], 156, [  ] ]
    [ [ 15, 48 ], 169, [  ] ]
    [ [ 15, 49 ], 199, [  ] ]
    [ [ 15, 50 ], 94, [  ] ]
    [ [ 15, 51 ], 169, [  ] ]
    [ [ 15, 52 ], 201, [  ] ]
    [ [ 15, 53 ], 456, [  ] ]
    [ [ 15, 54 ], 201, [  ] ]
    [ [ 15, 55 ], 201, [  ] ]
    [ [ 15, 56 ], 186, [  ] ]
    [ [ 15, 57 ], 124, [  ] ]
    [ [ 15, 58 ], 199, [  ] ]
    [ [ 15, 59 ], 124, [  ] ]
    [ [ 15, 60 ], 169, [  ] ]
    [ [ 15, 61 ], 471, [  ] ]
    [ [ 15, 62 ], 471, [  ] ]
    [ [ 15, 63 ], 456, [  ] ]
    [ [ 15, 64 ], 201, [  ] ]
    [ [ 15, 65 ], 199, [  ] ]
    [ [ 15, 66 ], 199, [  ] ]
    [ [ 15, 67 ], 124, [  ] ]
    [ [ 15, 68 ], 199, [  ] ]
    [ [ 15, 69 ], 456, [  ] ]
    [ [ 15, 70 ], 471, [  ] ]
    [ [ 15, 71 ], 111, [  ] ]
    [ [ 15, 72 ], 156, [  ] ]
    [ [ 15, 73 ], 199, [  ] ]
    [ [ 15, 74 ], 199, [  ] ]
    [ [ 15, 75 ], 124, [  ] ]
    [ [ 15, 76 ], 471, [  ] ]
    [ [ 15, 77 ], 471, [  ] ]
    [ [ 15, 78 ], 456, [  ] ]
    [ [ 15, 79 ], 201, [  ] ]
    [ [ 15, 80 ], 201, [  ] ]
    [ [ 15, 81 ], 111, [  ] ]
    [ [ 15, 82 ], 199, [  ] ]
    [ [ 15, 83 ], 471, [  ] ]
    [ [ 15, 84 ], 201, [  ] ]
    [ [ 15, 85 ], 201, [  ] ]
    [ [ 15, 86 ], 201, [  ] ]
    [ [ 15, 87 ], 201, [  ] ]
    [ [ 15, 88 ], 471, [  ] ]
    [ [ 15, 89 ], 471, [  ] ]
    [ [ 15, 90 ], 471, [  ] ]
    [ [ 15, 91 ], 471, [  ] ]
    [ [ 15, 92 ], 124, [  ] ]
    [ [ 15, 93 ], 471, [  ] ]
    [ [ 15, 94 ], 199, [  ] ]
    [ [ 15, 95 ], 124, [  ] ]
    [ [ 15, 96 ], 199, [  ] ]
    [ [ 15, 97 ], 199, [  ] ]
    [ [ 15, 98 ], 124, [  ] ]
    [ [ 15, 99 ], 199, [  ] ]
    [ [ 15, 100 ], 199, [  ] ]
    [ [ 15, 101 ], 124, [  ] ]
    [ [ 15, 102 ], 199, [  ] ]

######  Example 5.3 (Computation of $H^{1}(G,[J_{G/H}]^{fl})=0$ where
$G=12T295\simeq M_{12}$ and $G=22T38\simeq M_{22}$).

    gap> Read("FlabbyResolutionFromBase.gap");

    gap> G:=TransitiveGroup(12,295);
    M(12)
    gap> F:=FlabbyResolutionLowRankFromGroup(Norm1TorusJ(12,295),G).actionF;
    <matrix group with 2 generators>
    gap> [[12,295],Length(F.1),Filtered(H1(F),x->x>1)];
    [ [ 12, 295 ], 814, [  ] ]

    gap> G:=TransitiveGroup(22,38);
    t22n38
    gap> StructureDescription(G);
    "M22"
    gap> F:=FlabbyResolutionLowRankFromGroup(Norm1TorusJ(22,38),G).actionF;
    <matrix group with 2 generators>
    gap> [[22,38],Length(F.1),Filtered(H1(F),x->x>1)];
    [ [ 22, 38 ], 672, [  ] ]

##  6\. Proof of Theorem 1.18

Let $k$ be a number field, $K/k$ be a finite extension,
$\mathbbm{A}_{K}^{\times}$ be the idele group of $K$ and $L/k$ be the Galois
closure of $K/k$. Let $G={\rm Gal}(L/k)=nTm$ be a transitive subgroup of
$S_{n}$ and $H={\rm Gal}(L/K)$ with $[G:H]=n$.

For $x,y\in G$, we denote $[x,y]=x^{-1}y^{-1}xy$ the commutator of $x$ and
$y$, and $[G,G]$ the commutator group of $G$. Let $V_{k}$ be the set of all
places of $k$ and $G_{v}$ be the decomposition group of $G$ at $v\in V_{k}$.

######  Definition 6.1 (Drakokhrust and Platonov [PD85a, page 350], [DP87,
page 300]).

Let $k$ be a number field, $L\supset K\supset k$ be a tower of finite
extensions where $L$ is normal over $k$.

We call the group

$\displaystyle{\rm Obs}(K/k)=(N_{K/k}(\mathbbm{A}_{K}^{\times})\cap
k^{\times})/N_{K/k}(K^{\times})$

the total obstruction to the Hasse norm principle for $K/k$ and

$\displaystyle{\rm Obs}_{1}(L/K/k)=\left(N_{K/k}(\mathbbm{A}_{K}^{\times})\cap
k^{\times}\right)/\left((N_{L/k}(\mathbbm{A}_{L}^{\times})\cap
k^{\times})N_{K/k}(K^{\times})\right)$

the first obstruction to the Hasse norm principle for $K/k$ corresponding to
the tower $L\supset K\supset k$.

Note that (i) ${\rm Obs}(K/k)=1$ if and only if the Hasse norm principle holds
for $K/k$; and (ii) ${\rm Obs}_{1}(L/K/k)={\rm
Obs}(K/k)/(N_{L/k}(\mathbbm{A}_{L}^{\times})\cap k^{\times})$.

Drakokhrust and Platonov gave a formula for computing the first obstruction
${\rm Obs}_{1}(L/K/k)$:

######  Theorem 6.2 (Drakokhrust and Platonov [PD85a, page 350], [PD85b, pages
789–790], [DP87, Theorem 1]).

Let $k$ be a number field, $L\supset K\supset k$ be a tower of finite
extensions where $L$ is normal over $k$. Let $G={\rm Gal}(L/k)$ and $H={\rm
Gal}(L/K)$. Then

$\displaystyle{\rm Obs}_{1}(L/K/k)\simeq{\rm Ker}\,\psi_{1}/\varphi_{1}({\rm
Ker}\,\psi_{2})$

where

$\displaystyle\begin{CD}H/[H,H]@>{\psi_{1}}>{}>G/[G,G]\\\
@A{}A{\varphi_{1}}A@A{}A{\varphi_{2}}A\\\ \displaystyle{\bigoplus_{v\in
V_{k}}\left(\bigoplus_{w\mid
v}H_{w}/[H_{w},H_{w}]\right)}@>{\psi_{2}}>{}>\displaystyle{\bigoplus_{v\in
V_{k}}G_{v}/[G_{v},G_{v}]},\end{CD}$

$\psi_{1}$, $\varphi_{1}$ and $\varphi_{2}$ are defined by the inclusions
$H\subset G$, $H_{w}\subset H$ and $G_{v}\subset G$ respectively, and

$\displaystyle\psi_{2}(h[H_{w},H_{w}])=x^{-1}hx[G_{v},G_{v}]$

for $h\in H_{w}=H\cap x^{-1}hx[G_{v},G_{v}]$ $(x\in G)$.

Let $\psi_{2}^{v}$ be the restriction of $\psi_{2}$ to the subgroup
$\bigoplus_{w\mid v}H_{w}/[H_{w},H_{w}]$ with respect to $v\in V_{k}$ and
$\psi_{2}^{\rm nr}$ (resp. $\psi_{2}^{\rm r}$) be the restriction of
$\psi_{2}$ to the unramified (resp. the ramified) places $v$ of $k$.

######  Proposition 6.3 (Drakokhrust and Platonov [DP87]).

Let $k$, $L\supset K\supset k$, $G$ and $H$ be as in Theorem 6.2.
(i) $($[DP87, Lemma 1]$)$ Places $w_{i}\mid v$ of $K$ are in one-to-one
correspondence with the set of double cosets in the decomposition
$G=\cup_{i=1}^{r_{v}}Hx_{i}G_{v}$ where $H_{w_{i}}=H\cap
x_{i}G_{v}x_{i}^{-1}$;
(ii) $($[DP87, Lemma 2]$)$ If $G_{v_{1}}\leq G_{v_{2}}$, then
$\varphi_{1}({\rm Ker}\,\psi_{2}^{v_{1}})\subset\varphi_{1}({\rm
Ker}\,\psi_{2}^{v_{2}})$;
(iii) $($[DP87, Theorem 2]$)$ $\varphi_{1}({\rm Ker}\,\psi_{2}^{\rm
nr})=\Phi^{G}(H)/[H,H]$ where $\Phi^{G}(H)=\langle[h,x]\mid h\in H\cap
xHx^{-1},x\in G\rangle$;
(iv) $($[DP87, Lemma 8]$)$ If $[K:k]=p^{r}$ $(r\geq 1)$ and ${\rm
Obs}(K_{p}/k_{p})=1$ where $k_{p}=L^{G_{p}}$, $K_{p}=L^{H_{p}}$, $G_{p}$ and
$H_{p}\leq H\cap G_{p}$ are $p$-Sylow subgroups of $G$ and $H$ respectively,
then ${\rm Obs}(K/k)=1$.

######  Remark 6.4.

The inverse direction of Proposition 6.3 (iv) does not hold in general. For
example, if $n=8$, $G=8T13\simeq A_{4}\times C_{2}$ and there exists a place
$v$ of $k$ such that $G_{v}\simeq V_{4}$, then ${\rm Obs}(K/k)=1$ but
$G_{2}=8T3\simeq(C_{2})^{3}$ and ${\rm Obs}(K_{2}/k_{2})\neq 1$ may occur (see
Theorem 1.18 and Table $2$).

######  Theorem 6.5 (Drakokhrust and Platonov [DP87, Theorem 3, Corollary 1]).

Let $k$, $L\supset K\supset k$, $G$ and $H$ be as in Theorem 6.2. Let
$H_{i}\leq G_{i}\leq G$ $(1\leq i\leq m)$, $H_{i}\leq H\cap G_{i}$,
$k_{i}=L^{G_{i}}$ and $K_{i}=L^{H_{i}}$. If ${\rm Obs}(K_{i}/k_{i})=1$ for all
$1\leq i\leq m$ and

$\displaystyle\bigoplus_{i=1}^{m}\widehat{H}^{-3}(G_{i},\mathbbm{Z})\xrightarrow{\rm
cores}\widehat{H}^{-3}(G,\mathbbm{Z})$

is surjective, then ${\rm Obs}(K/k)={\rm Obs}_{1}(L/K/k)$. In particular, if
$[K:k]=n$ is square-free, then ${\rm Obs}(K/k)={\rm Obs}_{1}(L/K/k)$.

We note that if $L/k$ is an unramified extension, then $A(T)=0$ and
$H^{1}(G,[J_{G/H}]^{fl})\simeq\Sha(T)\simeq{\rm Obs}(K/k)$ where
$T=R^{(1)}_{K/k}(\mathbbm{G}_{m})$ (see Theorem 1.2 and Theorem 1.9). If, in
addition, ${\rm Obs}(K/k)={\rm Obs}_{1}(L/K/k)$ (e.g. $[K:k]=6,10,14,15$;
square-free, see Theorem 6.5), then ${\rm Obs}(K/k)={\rm Obs}_{1}(L/K/k)={\rm
Ker}\,\psi_{1}/\varphi_{1}({\rm Ker}\,\psi_{2}^{\rm nr})\simeq{\rm
Ker}\,\psi_{1}/(\Phi^{G}(H)/[H,H])$ (see Proposition 6.3 (iii)).

######  Theorem 6.6 (Drakokhrust [Dra89, Theorem 1], see also Opolka [Opo80,
Satz 3] for the existence of $\widetilde{L}$).

Let $k$, $L\supset K\supset k$, $G$ and $H$ be as in Theorem 6.2. Let
$\widetilde{L}\supset L\supset k$ be a tower of Galois extensions with
$\widetilde{G}={\rm Gal}(\widetilde{L}/k)$ and $\widetilde{H}={\rm
Gal}(\widetilde{L}/K)$ which correspond to a central extension $1\to
A\to\widetilde{G}\to G\to 1$ with $A\cap[\widetilde{G},\widetilde{G}]\simeq
M(G)$; the Schur multiplier of $G$ $($this is equivalent to the inflation
$M(\widetilde{G})\to M(G)$ is zero map, see Beyl and Tappe [BT82, Proposition
2.13, page 85]$)$. Then ${\rm Obs}(K/k)={\rm Obs}_{1}(\widetilde{L}/K/k)$. In
particular, if $\widetilde{G}$ is a Schur cover of $G$, i.e. $A\simeq M(G)$,
then ${\rm Obs}(K/k)={\rm Obs}_{1}(\widetilde{L}/K/k)$.

Indeed, Drakokhrust [Dra89, Theorem 1] shows that ${\rm Obs}(K/k)={\rm
Obs}_{1}(\widetilde{L}/K/k)/D\simeq{\rm
Ker}\,\widetilde{\psi}_{1}/\widetilde{\varphi}_{1}({\rm
Ker}\,\widetilde{\psi}_{2})$ and ${\rm Obs}(K/k)\simeq{\rm
Ker}\,\widetilde{\psi}_{1}/\widetilde{\varphi}_{1}({\rm
Ker}\,\widetilde{\psi}_{2})$ with some modified definitions of
$\widetilde{\varphi}_{1},\widetilde{\psi}_{2}$, i.e. $D=1$ (see also the
review of Colliot-Thélène in zbMATH: https://zbmath.org/?q=an:0596.12011).

We made the following functions of GAP ([GAP]) which will be used in the proof
of Theorem 1.18.

FirstObstructionN($G,H$).ker returns the list $[l_{1},[l_{2},l_{3}]]$ where
$l_{1}$ is the abelian invariant of the numerator of the first obstruction
${\rm Ker}\,\psi_{1}=\langle y_{1},\ldots,y_{t}\rangle$ with respect to $G$,
$H$ as in Theorem 6.2, $l_{2}=[e_{1},\ldots,e_{m}]$ is the abelian invariant
of $H^{ab}=H/[H,H]=\langle x_{1},\ldots,x_{m}\rangle$ with $e_{i}={\rm
order}(x_{i})$ and $l_{3}=[l_{3,1},\ldots,l_{3,t}]$,
$l_{3,i}=[r_{i,1},\ldots,r_{i,m}]$ is the list with
$y_{i}=x_{1}^{r_{i,1}}\cdots x_{m}^{r_{i,m}}$ for $H\leq G\leq S_{n}$.

FirstObstructionN($G$).ker returns the same as FirstObstructionN($G,H$).ker
where $H={\rm Stab}_{1}(G)$ is the stabilizer of $1$ in $G\leq S_{n}$.

FirstObstructionDnr($G,H$).Dnr returns the list $[l_{1},[l_{2},l_{3}]]$ where
$l_{1}$ is the abelian invariant of the unramified part of the denominator of
the first obstruction $\varphi_{1}({\rm Ker}\,\psi_{2}^{\rm
nr})=\Phi^{G}(H)/[H,H]=\langle y_{1},\ldots,y_{t}\rangle$ with respect to $G$,
$H$ as in Proposition 6.3 (iii), $l_{2}=[e_{1},\ldots,e_{m}]$ is the abelian
invariant of $H^{ab}=H/[H,H]=\langle x_{1},\ldots,x_{m}\rangle$ with
$e_{i}={\rm order}(x_{i})$ and $l_{3}=[l_{3,1},\ldots,l_{3,t}]$,
$l_{3,i}=[r_{i,1},\ldots,r_{i,m}]$ is the list with
$y_{i}=x_{1}^{r_{i,1}}\cdots x_{m}^{r_{i,m}}$ for $H\leq G\leq S_{n}$.

FirstObstructionDnr($G$).Dnr returns the same as
FirstObstructionDnr($G,H$).Dnr where $H={\rm Stab}_{1}(G)$ is the stabilizer
of $1$ in $G\leq S_{n}$.

FirstObstructionDr($G,G_{v},H$).Dr returns the list $[l_{1},[l_{2},l_{3}]]$
where $l_{1}$ is the abelian invariant of the ramified part of the denominator
of the first obstruction $\varphi_{1}({\rm Ker}\,\psi_{2}^{v})=\langle
y_{1},\ldots,y_{t}\rangle$ with respect to $G$, $G_{v}$, $H$ as in Theorem
6.2, $l_{2}=[e_{1},\ldots,e_{m}]$ is the abelian invariant of
$H^{ab}=H/[H,H]=\langle x_{1},\ldots,x_{m}\rangle$ with $e_{i}={\rm
order}(x_{i})$ and $l_{3}=[l_{3,1},\ldots,l_{3,t}]$,
$l_{3,i}=[r_{i,1},\ldots,r_{i,m}]$ is the list with
$y_{i}=x_{1}^{r_{i,1}}\cdots x_{m}^{r_{i,m}}$ for $G_{v},H\leq G\leq S_{n}$.

FirstObstructionDr($G,G_{v}$).Dr returns the same as
FirstObstructionDr($G,G_{v},H$).Dr where $H={\rm Stab}_{1}(G)$ is the
stabilizer of $1$ in $G\leq S_{n}$.

SchurCoverG($G$).SchurCover (resp. SchurCoverG($G$).epi) returns one of the
Schur covers $\widetilde{G}$ of $G$ (resp. the surjective map $\pi$) in a
central extension $1\to A\to\widetilde{G}\xrightarrow{\pi}G\to 1$ with
$A\simeq M(G)$; Schur multiplier of $G$ (see Karpilovsky [Kap87, page 16]).
The Schur covers $\widetilde{G}$ are stem extensions, i.e. $A\leq
Z(\widetilde{G})\cap[\widetilde{G},\widetilde{G}]$, of the maximal size. This
function is based on the built-in function EpimorphismSchurCover in GAP.

MinimalStemExtensions($G$)[$j$].MinimalStemExtension (resp.
MinimalStemExtensions($G$)[$j$].epi) returns the $j$-th minimal stem extension
$\overline{G}=\widetilde{G}/A^{\prime}$, i.e. $\overline{A}\leq
Z(\overline{G})\cap[\overline{G},\overline{G}]$, of $G$ provided by the Schur
cover $\widetilde{G}$ of $G$ via SchurCoverG($G$).SchurCover where
$A^{\prime}$ is the $j$-th maximal subgroup of $A=M(G)$ (resp. the surjective
map $\overline{\pi}$) in the commutative diagram

$\displaystyle\begin{CD}1@>{}>{}>A=M(G)@>{}>{}>\widetilde{G}@>{\pi}>{}>G@>{}>{}>1\\\
@V{}V{}V@V{}V{}V\Big{\|}\\\
1@>{}>{}>\overline{A}=A/A^{\prime}@>{}>{}>\overline{G}=\widetilde{G}/A^{\prime}@>{\overline{\pi}}>{}>G@>{}>{}>1\end{CD}$

(see Robinson [Rob96, Exercises 11.4]). This function is based on the built-in
function EpimorphismSchurCover in GAP.

ResolutionNormalSeries(LowerCentralSeries($G$), $n+1$) (resp.
ResolutionNormalSeries(DerivedSeries ($G$),$n+1$),
ResolutionFiniteGroup($G$,$n+1$)) returns a free resolution $RG$ of $G$ when
$G$ is nilpotent (resp. solvable, finite). This function is the built-in
function of HAP ([HAP]) in GAP ([GAP]) .

ResHnZ($RG,RH,n$).HnGZ (resp. ResHnZ($RG,RH,n$).HnHZ) returns the abelian
invariants of $H^{n}(G,\mathbbm{Z})$ (resp. $H^{n}(H,\mathbbm{Z})$) with
respect to Smith normal form, for free resolutions $RG$ and $RH$ of $G$ and
$H$ respectively.

ResHnZ($RG,RH,n$).Res returns the list $L=[l_{1},\ldots,l_{s}]$ where
$H^{n}(G,\mathbbm{Z})=\langle x_{1},\ldots,x_{s}\rangle\xrightarrow{\rm
res}H^{n}(H,\mathbbm{Z})=\langle y_{1},\ldots,y_{t}\rangle$, ${\rm
res}(x_{i})=\prod_{j=1}^{t}y_{j}^{l_{i,j}}$ and
$l_{i}=[l_{i,1},\ldots,l_{i,t}]$ for free resolutions $RG$ and $RH$ of $G$ and
$H$ respectively.

ResHnZ($RG,RH,n$).Ker returns the list $L=[l_{1},[l_{2},l_{3}]]$ where $l_{1}$
is the abelian invariant of ${\rm Ker}\\{H^{n}(G,\mathbbm{Z})$
$\xrightarrow{\rm res}$ $H^{n}(H,\mathbbm{Z})\\}=\langle
y_{1},\ldots,y_{t}\rangle$, $l_{2}=[d_{1},\ldots,d_{s}]$ is the abelian
invariant of $H^{n}(G,\mathbbm{Z})=\langle x_{1},\ldots,x_{s}\rangle$ with
$d_{i}={\rm ord}(x_{i})$ and $l_{3}=[l_{3,1},\ldots,l_{3,t}]$,
$l_{3,j}=[r_{j,1},\ldots,r_{j,s}]$ is the list with
$y_{j}=x_{1}^{r_{j,1}}\cdots x_{s}^{r_{j,s}}$ for free resolutions $RG$ and
$RH$ of $G$ and $H$ respectively.

ResHnZ($RG,RH,n$).Coker returns the list $L=[l_{1},[l_{2},l_{3}]]$ where
$l_{1}=[e_{1},\ldots,e_{t}]$ is the abelian invariant of ${\rm
Coker}\\{H^{n}(G,\mathbbm{Z})$ $\xrightarrow{\rm res}$
$H^{n}(H,\mathbbm{Z})\\}=\langle\overline{y_{1}},\ldots,\overline{y_{t}}\rangle$
with $e_{j}={\rm ord}(\overline{y_{j}})$, $l_{2}=[d_{1},\ldots,d_{s}]$ is the
abelian invariant of $H^{n}(H,\mathbbm{Z})=\langle x_{1},\ldots,x_{s}\rangle$
with $d_{i}={\rm ord}(x_{i})$ and $l_{3}=[l_{3,1},\ldots,l_{3,t}]$,
$l_{3,j}=[r_{j,1},\ldots,r_{j,s}]$ is the list with
$\overline{y_{j}}=\overline{x_{1}}^{r_{j,1}}\cdots\overline{x_{s}}^{r_{j,s}}$
for free resolutions $RG$ and $RH$ of $G$ and $H$ respectively.

KerResH3Z($G,H$) returns the list $L=[l_{1},[l_{2},l_{3}]]$ where $l_{1}$ is
the abelian invariant of ${\rm Ker}\\{H^{3}(G,\mathbbm{Z})\xrightarrow{\rm
res}\oplus_{i=1}^{m^{\prime}}H^{3}(G_{i},\mathbbm{Z})\\}=\langle
y_{1},\ldots,y_{t}\rangle$ where $H_{i}\leq G_{i}\leq G$, $H_{i}\leq H\cap
G_{i}$, $[G_{i}:H_{i}]=n$ and the action of $G_{i}$ on
$\mathbbm{Z}[G_{i}/H_{i}]$ may be regarded as $nTm$ $(n\leq 15,n\neq 12)$
which is not in Table $1$, $l_{2}=[d_{1},\ldots,d_{s}]$ is the abelian
invariant of $H^{3}(G,\mathbbm{Z})=\langle x_{1},\ldots,x_{s}\rangle$ with
$d_{i^{\prime}}={\rm ord}(x_{i^{\prime}})$ and
$l_{3}=[l_{3,1},\ldots,l_{3,t}]$, $l_{3,j}=[r_{j,1},\ldots,r_{j,s}]$ is the
list with $y_{j}=x_{1}^{r_{j,1}}\cdots x_{s}^{r_{j,s}}$ for groups $G$ and $H$
(cf. Theorem 6.6).

The functions above are available from
https://www.math.kyoto-u.ac.jp/~yamasaki/Algorithm/Norm1ToriHNP.

Proof of Theorem 1.18.

Let $G={\rm Gal}(L/k)=nTm\leq S_{n}$ be the $m$-th transitive subgroup of
$S_{n}$ and $H={\rm Gal}(L/K)\leq G$ with $[G:H]=n$. Let $V_{k}$ be the set of
all places of $k$ and $G_{v}$ be the decomposition group of $G$ at $v\in
V_{k}$.

We split the proof into the following cases:
(1) $G=8Tm$ $(m=2,3,4,13,14,21,31,37,38)$,
(2) $G=8Tm$ $(m=9,11,15,19,22,32)$,
(3) $G=9Tm$ $(m=2,5,7,9,11,14,23)$,
(4) $G=10Tm$ $(m=7,26,32)$,
(5) $G=14T30$,
(6) $G=15Tm$ $(m=9,14)$.

For the reader’s convenience, we also give the GAP computations to the known
cases: $G=4T2\simeq V_{4},4T4\simeq A_{4},6T4\simeq A_{4},6T12\simeq A_{5}$
(see Example 6.9 and Example 6.10).

In order to prove the statement of the theorem, we may assume that $H={\rm
Stab}_{1}(G)$ is the stabilizer of $1$ in $G$, i.e.
$L=k(\theta_{1},\ldots,\theta_{n})$ and $K=L^{H}=k(\theta_{1})$, without loss
of generality except for the cases (2) $G=8Tm$ $(m=9,11,15,19,22,32)$ and
$G=10T32\simeq S_{6}$ because the center $Z(G)$ and the commutator group
$[G^{\prime},G^{\prime}]$ where $G^{\prime}\leq G$ is a characteristic
subgroup in the statement of the theorem, are characteristic subgroups of $G$,
i.e. invariants under the automorphisms of $G$.

For the cases (2) $G=8Tm$, by the assumption of the statement of the theorem,
we may assume that $H={\rm Stab}_{1}(G)$ is the stabilizer of $1$ in $G$
because (the multi-set) $\\{{\rm Orb}_{G^{\prime}}(i)\mid 1\leq i\leq n\\}$
$(G^{\prime}\leq G)$ is invariant under the conjugacy actions of $G$, i.e.
inner automorphisms of $G$.

For the case $G=10T32\simeq S_{6}$, there exist exactly $10$ subgroups $H\leq
G$ with $[G:H]=10$ which are conjugate in $G$. Hence we may assume that
$H={\rm Stab}_{1}(G)$ without loss of generality.

By Theorem 1.2 and Theorem 1.15, it is enough to give a necessary and
sufficient condition for $\Sha(T)=0$.

(1) $n=8$: $G=8Tm$ $(m=2,3,4,13,14,21,31,37,38)$. Applying the functions
FirstObstructionN($G$) and FirstObstructionDnr($G$), we have ${\rm
Obs}_{1}(L/K/k)=1$ except for $G=8T21\simeq(C_{2})^{3}\rtimes C_{4}$. For
$G=8T21$, we obtain that ${\rm Obs}_{1}(L/K/k)\simeq\mathbbm{Z}/2\mathbbm{Z}$.

(1-1) The case $G=8T3\simeq(C_{2})^{3}$. This case follows from Theorem 1.8
because $H=1$. See also Example 6.11 and the second paragraph after Theorem
1.8.

(1-2) The case $G=8T21\simeq(C_{2})^{3}\rtimes C_{4}$. We have $H=H^{ab}\simeq
C_{2}\times C_{2}$. Applying FirstObstructionN($G$) and
FirstObstructionDnr($G$), we obtain that ${\rm Ker}\,\psi_{1}/\varphi_{1}({\rm
Ker}\,\psi_{2}^{\rm nr})$ $\simeq$ $\mathbbm{Z}/2\mathbbm{Z}$. By Theorem
1.15, we get ${\rm Obs}(K/k)={\rm Obs}_{1}(L/K/k)$ when $L/k$ is unramified
(see the paragraph after Theorem 6.5). Use Theorem 6.5. Applying the function
KerResH3Z(G,H), we see that ${\rm Ker}\\{H^{3}(G,\mathbbm{Z})\xrightarrow{\rm
res}\oplus_{i=1}^{m^{\prime}}H^{3}(G_{i},\mathbbm{Z})\\}=0$ and hence
$\oplus_{i=1}^{m^{\prime}}\widehat{H}^{-3}(G_{i},\mathbbm{Z})\xrightarrow{\rm
cores}\widehat{H}^{-3}(G,\mathbbm{Z})$ is surjective. It follows from Theorem
6.5 that ${\rm Obs}(K/k)={\rm Obs}_{1}(L/K/k)$. Applying the function
FirstObstructionDr($G,G^{\prime}$) for all subgroups $G^{\prime}\leq G$, we
find that ${\rm Obs}_{1}(L/K/k)$ $=$ $1$ if and only if there exists $v\in
V_{k}$ such that $G_{v}=G$ (see Example 6.11).

(1-3) The case $G=8Tm$ $(m=2,4,13,14,37)$. Because ${\rm Obs}_{1}(K/k)=1$, we
just apply Theorem 6.6. We have the Schur multiplier
$M(G)\simeq\mathbbm{Z}/2\mathbbm{Z}$ for $G=8Tm$ $(m=2,4,13,14,37)$.

(1-3-1) The case $G=8T2\simeq C_{4}\times C_{2}$ (see also Theorem 1.8 because
$H=1$). Apply Theorem 6.6. We obtain a Schur cover $1\to
M(G)\simeq\mathbbm{Z}/2\mathbbm{Z}\to\widetilde{G}\xrightarrow{\pi}G\to 1$
with $\widetilde{G}\simeq(C_{4}\times C_{2})\rtimes C_{2}$,
$\widetilde{H}\simeq C_{2}$ and ${\rm Obs}(K/k)={\rm
Obs}_{1}(\widetilde{L}/K/k)$. By Theorem 1.15, ${\rm
Ker}\,\widetilde{\psi}_{1}/\widetilde{\varphi}_{1}({\rm
Ker}\,\widetilde{\psi}_{2}^{\rm nr})\simeq\mathbbm{Z}/2\mathbbm{Z}$ (see the
paragraph after Theorem 6.5). By applying
FirstObstructionDr($\widetilde{G},\widetilde{G}^{\prime},\widetilde{H}$) for
all subgroups $\widetilde{G}^{\prime}\leq\widetilde{G}$, we obtain that ${\rm
Obs}_{1}(\widetilde{L}/K/k)=1$ if and only if there exists $v\in V_{k}$ such
that $\widetilde{G}_{v}=\widetilde{G}$ if and only if there exists $v\in
V_{k}$ such that $G_{v}=G$ (see Example 6.11).

(1-3-2) The case $G=8T4\simeq D_{4}$ (see also Theorem 1.8 because $H=1$).
Apply Theorem 6.6. We obtain a Schur cover $1\to
M(G)\simeq\mathbbm{Z}/2\mathbbm{Z}\to\widetilde{G}\xrightarrow{\pi}G\to 1$
with $\widetilde{G}\simeq D_{8}$, $\widetilde{H}\simeq C_{2}$ and ${\rm
Obs}(K/k)={\rm Obs}_{1}(\widetilde{L}/K/k)$. By Theorem 1.15, ${\rm
Ker}\,\widetilde{\psi}_{1}/\widetilde{\varphi}_{1}({\rm
Ker}\,\widetilde{\psi}_{2}^{\rm nr})\simeq\mathbbm{Z}/2\mathbbm{Z}$. By
applying
FirstObstructionDr($\widetilde{G},\widetilde{G}^{\prime},\widetilde{H}$) for
all subgroups $\widetilde{G}^{\prime}\leq\widetilde{G}$, we obtain that ${\rm
Obs}_{1}(\widetilde{L}/K/k)=1$ if and only if there exists $v\in V_{k}$ such
that $D_{4}\leq\widetilde{G}_{v}$ if and only if there exists $v\in V_{k}$
such that $V_{4}\leq G_{v}$ (see Example 6.11).

(1-3-3) The case $G=8T13\simeq A_{4}\times C_{2}$. We have $H\simeq C_{3}$.
Apply Theorem 6.6. We obtain a Schur cover $1\to
M(G)\simeq\mathbbm{Z}/2\mathbbm{Z}\to\widetilde{G}\xrightarrow{\pi}G\to 1$
with $\widetilde{G}\simeq((C_{4}\times C_{2})\rtimes C_{2})\rtimes C_{3}$,
$\widetilde{H}\simeq C_{6}$ and ${\rm Obs}(K/k)={\rm
Obs}_{1}(\widetilde{L}/K/k)$. By Theorem 1.15, ${\rm
Ker}\,\widetilde{\psi}_{1}/\widetilde{\varphi}_{1}({\rm
Ker}\,\widetilde{\psi}_{2}^{\rm nr})\simeq\mathbbm{Z}/2\mathbbm{Z}$. By
applying
FirstObstructionDr($\widetilde{G},\widetilde{G}^{\prime},\widetilde{H}$) for
all subgroups $\widetilde{G}^{\prime}\leq\widetilde{G}$, we obtain that ${\rm
Obs}_{1}(\widetilde{L}/K/k)=1$ if and only if there exists $v\in V_{k}$ such
that $V_{4}\leq G_{v}$ (see Example 6.11).

(1-3-4) The case $G=8T14\simeq S_{4}$. We have $H\simeq C_{3}$. Apply Theorem
6.6. We obtain a Schur cover $1\to
M(G)\simeq\mathbbm{Z}/2\mathbbm{Z}\to\widetilde{G}\xrightarrow{\pi}G\to 1$
with $\widetilde{G}\simeq GL_{2}(\mathbbm{F}_{3})$ and $\widetilde{H}\simeq
C_{6}$. By Theorem 1.15, ${\rm
Ker}\,\widetilde{\psi}_{1}/\widetilde{\varphi}_{1}({\rm
Ker}\,\widetilde{\psi}_{2}^{\rm nr})\simeq\mathbbm{Z}/2\mathbbm{Z}$. By
applying
FirstObstructionDr($\widetilde{G},\widetilde{G}^{\prime},\widetilde{H}$) for
all subgroups $\widetilde{G}^{\prime}\leq\widetilde{G}$, we obtain that ${\rm
Obs}_{1}(\widetilde{L}/K/k)=1$ if and only if there exists $v\in V_{k}$ such
that $V_{4}\leq G_{v}$ (see Example 6.11).

(1-3-5) The case $G=8T37\simeq{\rm PSL}_{3}(\mathbbm{F}_{2})\simeq{\rm
PSL}_{2}(\mathbbm{F}_{7})$. We have $H\simeq C_{7}\rtimes C_{3}$. We obtain a
Schur cover $1\to
M(G)\simeq\mathbbm{Z}/2\mathbbm{Z}\to\widetilde{G}\xrightarrow{\pi}G\to 1$
with $\widetilde{G}\simeq{\rm SL}_{2}(\mathbbm{F}_{7})$, $\widetilde{H}\simeq
C_{2}\times(C_{7}\rtimes C_{3})$ and ${\rm Obs}(K/k)={\rm
Obs}_{1}(\widetilde{L}/K/k)$. By Theorem 1.15, ${\rm
Ker}\,\widetilde{\psi}_{1}/\widetilde{\varphi}_{1}({\rm
Ker}\,\widetilde{\psi}_{2}^{\rm nr})\simeq\mathbbm{Z}/2\mathbbm{Z}$. By
applying
FirstObstructionDr($\widetilde{G},\widetilde{G}^{\prime},\widetilde{H}$) for
all subgroups $\widetilde{G}^{\prime}\leq\widetilde{G}$, we obtain that ${\rm
Obs}_{1}(\widetilde{L}/K/k)=1$ if and only if there exists $v\in V_{k}$ such
that $V_{4}\leq G_{v}$ (see Example 6.11).

(1-4) The case $G=8Tm$ $(m=31,38)$. Applying FirstObstructionN($G$) and
FirstObstructionDnr($G$), we have ${\rm Obs}_{1}(L/K/k)=1$. For $G=8T31$
(resp. $G=8T38$), we have $M(G)\simeq(\mathbbm{Z}/2\mathbbm{Z})^{\oplus 4}$
(resp. $M(G)\simeq(\mathbbm{Z}/2\mathbbm{Z})^{\oplus 2}$). Hence we take a
minimal stem extension $\overline{G}=\widetilde{G}/A^{\prime}$, i.e.
$\overline{A}\leq Z(\overline{G})\cap[\overline{G},\overline{G}]$, of $G$ in
the commutative diagram

$\displaystyle\begin{CD}1@>{}>{}>A=M(G)@>{}>{}>\widetilde{G}@>{\pi}>{}>G@>{}>{}>1\\\
@V{}V{}V@V{}V{}V\Big{\|}\\\
1@>{}>{}>\overline{A}=A/A^{\prime}@>{}>{}>\overline{G}=\widetilde{G}/A^{\prime}@>{\overline{\pi}}>{}>G@>{}>{}>1\end{CD}$

with $\overline{A}\simeq\mathbbm{Z}/2\mathbbm{Z}$ via the function
MinimalStemExtensions($G$)[$j$].MinimalStemExtension. Then we apply Theorem
6.5 instead of Theorem 6.6.

(1-4-1) The case $G=8T31\simeq((C_{2})^{4}\rtimes C_{2})\rtimes C_{2}$. We
have $H\simeq(C_{2})^{3}$ and $M(G)\simeq(\mathbbm{Z}/2\mathbbm{Z})^{\oplus
4}$. Applying the function
MinimalStemExtensions($G$)[$j$].MinimalStemExtension, we get the minimal stem
extensions $\overline{G}_{1},\ldots,\overline{G}_{15}$ of $G$. Use Theorem
6.5. Applying KerResH3Z(G,H), we see that ${\rm
Ker}\\{H^{3}(\overline{G}_{1},\mathbbm{Z})\xrightarrow{\rm
res}\oplus_{i=1}^{m^{\prime}}H^{3}(G_{i},\mathbbm{Z})\\}=0$ but ${\rm
Ker}\\{H^{3}(\overline{G}_{j},\mathbbm{Z})\xrightarrow{\rm
res}\oplus_{i=1}^{m^{\prime}}H^{3}(G_{i},\mathbbm{Z})\\}\simeq\mathbbm{Z}/2\mathbbm{Z}$
for $j\in J:=\\{j\mid 2\leq j\leq 15\\}$. Because
$\oplus_{i=1}^{m^{\prime}}\widehat{H}^{-3}(G_{i},\mathbbm{Z})\xrightarrow{\rm
cores}\widehat{H}^{-3}(\overline{G}_{1},\mathbbm{Z})$ is surjective, it
follows from Theorem 6.5 that ${\rm Obs}(K/k)={\rm
Obs}_{1}(\overline{L}_{1}/K/k)$. We also checked that ${\rm
Ker}\,\overline{\psi}_{1}/\overline{\varphi}_{1}({\rm
Ker}\,\overline{\psi}_{1}^{\rm nr})$ $\simeq$ $\mathbbm{Z}/2\mathbbm{Z}$ for
$\overline{G}_{1}$ and ${\rm
Ker}\,\overline{\psi}_{1}/\overline{\varphi}_{1}({\rm
Ker}\,\overline{\psi}_{2}^{\rm nr})=0$ for $\overline{G}_{j}$ $(j\in J)$. This
implies that ${\rm Obs}(K/k)\neq{\rm Obs}_{1}(\overline{L}_{j}/K/k)$ when
$\overline{L}_{j}/k$ is unramified for $j\in J$. Apply
FirstObstructionDr($\overline{G}_{1},\overline{G}_{1}^{\prime},\overline{H}_{1}$)
for all subgroups $\overline{G}_{1}^{\prime}\leq\overline{G}_{1}$. We find
that ${\rm Obs}_{1}(\overline{L}_{1}/K/k)$ $=$ $1$ if and only if there exists
$v\in V_{k}$ such that (i) $V_{4}\leq G_{v}$ where $V_{4}\cap[G,G]=1$
$($equivalently, $|{\rm Orb}_{V_{4}}(i)|=4$ for any $1\leq i\leq 8$ and
$V_{4}\cap Z(G)=1$$)$, (ii) $C_{4}\times C_{2}\leq G_{v}$ where $(C_{4}\times
C_{2})\cap[G,G]\simeq C_{2}$ $($equivalently, $C_{4}\times C_{2}$ is
transitive in $S_{8}$$)$ or (iii) $(C_{2})^{3}\rtimes C_{4}\leq G_{v}$ (see
Details 6.8 and Example 6.11).

(1-4-2) The case $G=8T38\simeq(((C_{2})^{4}\rtimes C_{2})\rtimes C_{2})\rtimes
C_{3}$. We have $H\simeq C_{2}\times A_{4}$ and
$M(G)\simeq(\mathbbm{Z}/2\mathbbm{Z})^{\oplus 2}$. Applying the function
MinimalStemExtensions($G$)[$j$].MinimalStemExtension, we get the minimal stem
extensions $\overline{G}_{1},\overline{G}_{2},\overline{G}_{3}$ of $G$. Use
Theorem 6.5. Applying KerResH3Z(G,H), we see that ${\rm
Ker}\\{H^{3}(\overline{G}_{2},\mathbbm{Z})\xrightarrow{\rm
res}\oplus_{i=1}^{m^{\prime}}H^{3}(G_{i},\mathbbm{Z})\\}=0$ but ${\rm
Ker}\\{H^{3}(\overline{G}_{j},\mathbbm{Z})\xrightarrow{\rm
res}\oplus_{i=1}^{m^{\prime}}H^{3}(G_{i},\mathbbm{Z})\\}\simeq\mathbbm{Z}/2\mathbbm{Z}$
for $j\in J:=\\{1,3\\}$. We have that
$\oplus_{i=1}^{m^{\prime}}\widehat{H}^{-3}(G_{i},\mathbbm{Z})\xrightarrow{\rm
cores}\widehat{H}^{-3}(\overline{G}_{2},\mathbbm{Z})$ is surjective. By
Theorem 6.5, ${\rm Obs}(K/k)={\rm Obs}_{1}(\overline{L}_{2}/K/k)$. We also
checked that ${\rm Ker}\,\overline{\psi}_{1}/\overline{\varphi}_{1}({\rm
Ker}\,\overline{\psi}_{1}^{\rm nr})$ $\simeq$ $\mathbbm{Z}/2\mathbbm{Z}$ for
$\overline{G}_{2}$ and ${\rm
Ker}\,\overline{\psi}_{1}/\overline{\varphi}_{1}({\rm
Ker}\,\overline{\psi}_{2}^{\rm nr})=0$ for $\overline{G}_{j}$ $(j\in J)$. This
implies that ${\rm Obs}(K/k)\neq{\rm Obs}_{1}(\overline{L}_{j}/K/k)$ when
$\overline{L}_{j}/k$ is unramified for $j\in J$. Apply
FirstObstructionDr($\overline{G}_{2},\overline{G}_{2}^{\prime},\overline{H}_{2}$)
for all subgroups $\overline{G}_{2}^{\prime}\leq\overline{G}_{2}$. We find
that ${\rm Obs}_{1}(\overline{L}_{2}/K/k)$ $=$ $1$ if and only if there exists
$v\in V_{k}$ such that (i) $V_{4}\leq G_{v}$ where $V_{4}\cap[{\rm
Syl}_{2}(G),{\rm Syl}_{2}(G)]=1$ with ${\rm Syl}_{2}(G)\lhd G$
$($equivalently, $|{\rm Orb}_{V_{4}}(i)|=4$ for any $1\leq i\leq 8$ and
$V_{4}\cap Z(G)=1$$)$, (ii) $C_{4}\times C_{2}\leq G_{v}$ where $(C_{4}\times
C_{2})\cap[{\rm Syl}_{2}(G),{\rm Syl}_{2}(G)]\simeq C_{2}$ $($equivalently,
$C_{4}\times C_{2}$ is transitive in $S_{8}$$)$ or (iii) $(C_{2})^{3}\rtimes
C_{4}\leq G_{v}$ (see Details 6.8 and Example 6.11).

(2) $n=8$: $G=8Tm$ $(m=9,11,15,19,22,32)$. We assume that $H={\rm
Stab}_{1}(G)$ by the assumption. Applying FirstObstructionN($G$), we have
${\rm Obs}_{1}(L/K/k)=1$. For the cases $G=8Tm$ $(m=9,11,15,19,22,32)$, we
also find that $(\mathbbm{Z}/2\mathbbm{Z})^{\oplus 2}\leq
M(G)\leq(\mathbbm{Z}/2\mathbbm{Z})^{\oplus 5}$. Hence we take a minimal stem
extension $\overline{G}=\widetilde{G}/A^{\prime}$, i.e. $\overline{A}\leq
Z(\overline{G})\cap[\overline{G},\overline{G}]$, of $G$ in the commutative
diagram

$\displaystyle\begin{CD}1@>{}>{}>A=M(G)@>{}>{}>\widetilde{G}@>{\pi}>{}>G@>{}>{}>1\\\
@V{}V{}V@V{}V{}V\Big{\|}\\\
1@>{}>{}>\overline{A}=A/A^{\prime}@>{}>{}>\overline{G}=\widetilde{G}/A^{\prime}@>{\overline{\pi}}>{}>G@>{}>{}>1\end{CD}$

with $\overline{A}\simeq\mathbbm{Z}/2\mathbbm{Z}$ via the function
MinimalStemExtensions($G$)[$j$].MinimalStemExtension. Then we apply Theorem
6.5 instead of Theorem 6.6 as in the case (1-4).

(2-1) The case $G=8T9\simeq D_{4}\times C_{2}$. We have $H\simeq C_{2}$. We
obtain that the Schur multiplier $M(G)\simeq(\mathbbm{Z}/2\mathbbm{Z})^{\oplus
3}$. By applying MinimalStemExtensions($G$)[$j$].MinimalStemExtension, we
obtain the minimal stem extensions $\overline{G}_{1},\ldots,\overline{G}_{7}$
of $G$. Use Theorem 6.5. Applying KerResH3Z(G,H), we see that ${\rm
Ker}\\{H^{3}(\overline{G}_{2},\mathbbm{Z})\xrightarrow{\rm
res}\oplus_{i=1}^{m^{\prime}}H^{3}(G_{i},\mathbbm{Z})\\}=0$ but ${\rm
Ker}\\{H^{3}(\overline{G}_{j},\mathbbm{Z})\xrightarrow{\rm
res}\oplus_{i=1}^{m^{\prime}}H^{3}(G_{i},\mathbbm{Z})\\}=\mathbbm{Z}/2\mathbbm{Z}$
for $j\in J:=\\{1,3,4,5,6,7\\}$. Because
$\oplus_{i=1}^{m^{\prime}}\widehat{H}^{-3}(G_{i},\mathbbm{Z})\xrightarrow{\rm
cores}\widehat{H}^{-3}(\overline{G}_{2},\mathbbm{Z})$ is surjective, it
follows from Theorem 6.5 that ${\rm Obs}(K/k)={\rm
Obs}_{1}(\overline{L}_{2}/K/k)$. We also checked that ${\rm
Ker}\,\overline{\psi}_{1}/\overline{\varphi}_{1}({\rm
Ker}\,\overline{\psi}_{2}^{\rm nr})\simeq\mathbbm{Z}/2\mathbbm{Z}$ for
$\overline{G}_{2}$ and ${\rm
Ker}\,\overline{\psi}_{1}/\overline{\varphi}_{1}({\rm
Ker}\,\overline{\psi}_{2}^{\rm nr})=0$ for $\overline{G}_{j}$ $(j\in J)$.
Hence ${\rm Obs}(K/k)\neq{\rm Obs}_{1}(\overline{L}_{j}/K/k)$ when
$\overline{L}_{j}/k$ is unramified for $j\in J$. Apply
FirstObstructionDr($\overline{G}_{2},\overline{G}_{2}^{\prime},\overline{H}_{2}$)
for all subgroups $\overline{G}_{2}^{\prime}\leq\overline{G}_{2}$. We find
that ${\rm Obs}_{1}(\overline{L}_{2}/K/k)$ $=$ $1$ if and only if there exists
$v\in V_{k}$ such that (i) $V_{4}\leq G_{v}$ where $|{\rm Orb}_{V_{4}}(i)|=4$
for any $1\leq i\leq 8$ and $V_{4}\cap[G,G]=1$; or (ii) $C_{4}\times C_{2}\leq
G_{v}$ (see Details 6.8 and Example 6.12).

(2-2) The case $G=8T11\simeq(C_{4}\times C_{2})\rtimes C_{2}$. We have
$H\simeq C_{2}$ and $M(G)\simeq(\mathbbm{Z}/2\mathbbm{Z})^{\oplus 3}$.
Applying MinimalStemExtensions($G$)[$j$].MinimalStemExtension. We get the
minimal stem extensions $\overline{G}_{1},\overline{G}_{2},\overline{G}_{3}$
of $G$. Use Theorem 6.5. Applying KerResH3Z(G,H), we see that ${\rm
Ker}\\{H^{3}(\overline{G}_{2},\mathbbm{Z})\xrightarrow{\rm
res}\oplus_{i=1}^{m^{\prime}}H^{3}(G_{i},\mathbbm{Z})\\}=0$ but ${\rm
Ker}\\{H^{3}(\overline{G}_{j},\mathbbm{Z})\xrightarrow{\rm
res}\oplus_{i=1}^{m^{\prime}}H^{3}(G_{i},\mathbbm{Z})\\}=\mathbbm{Z}/2\mathbbm{Z}$
for $j\in J:=\\{1,3\\}$. Because
$\oplus_{i=1}^{m^{\prime}}\widehat{H}^{-3}(G_{i},\mathbbm{Z})\xrightarrow{\rm
cores}\widehat{H}^{-3}(\overline{G}_{2},\mathbbm{Z})$ is surjective, it
follows from Theorem 6.5 that ${\rm Obs}(K/k)={\rm
Obs}_{1}(\overline{L}_{2}/K/k)$. We also checked that ${\rm
Ker}\,\overline{\psi}_{1}/\overline{\varphi}_{1}({\rm
Ker}\,\overline{\psi}_{2}^{\rm nr})$ $\simeq$ $\mathbbm{Z}/2\mathbbm{Z}$ for
$\overline{G}_{2}$ and ${\rm
Ker}\,\overline{\psi}_{1}/\overline{\varphi}_{1}({\rm
Ker}\,\overline{\psi}_{2}^{\rm nr})=0$ for $\overline{G}_{j}$ $(j\in J)$. This
implies that ${\rm Obs}(K/k)\neq{\rm Obs}_{1}(\overline{L}_{j}/K/k)$ when
$\overline{L}_{j}/k$ is unramified for $j\in J$. Apply
FirstObstructionDr($\overline{G}_{2},\overline{G}_{2}^{\prime},\overline{H}_{2}$)
for all subgroups $\overline{G}_{2}^{\prime}\leq\overline{G}_{2}$. We find
that ${\rm Obs}_{1}(\overline{L}_{2}/K/k)$ $=$ $1$ if and only if there exists
$v\in V_{k}$ such that $C_{4}\times C_{2}\leq G_{v}$ where $C_{4}\times C_{2}$
is transitive in $S_{8}$ (see Details 6.8 and Example 6.12).

(2-3) The case $G=8T15\simeq C_{8}\rtimes V_{4}$. We have $H\simeq V_{4}$ and
$M(G)\simeq(\mathbbm{Z}/2\mathbbm{Z})^{\oplus 2}$. Applying the function
MinimalStemExtensions($G$)[$j$].MinimalStemExtension, we get the minimal stem
extensions $\overline{G}_{1},\overline{G}_{2},\overline{G}_{3}$ of $G$. Use
Theorem 6.5. Applying KerResH3Z(G,H), we see that ${\rm
Ker}\\{H^{3}(\overline{G}_{1},\mathbbm{Z})\xrightarrow{\rm
res}\oplus_{i=1}^{m^{\prime}}H^{3}(G_{i},\mathbbm{Z})\\}=0$ but ${\rm
Ker}\\{H^{3}(\overline{G}_{j},\mathbbm{Z})\xrightarrow{\rm
res}\oplus_{i=1}^{m^{\prime}}H^{3}(G_{i},\mathbbm{Z})\\}=\mathbbm{Z}/2\mathbbm{Z}$
for $j\in J:=\\{2,3\\}$. Because
$\oplus_{i=1}^{m^{\prime}}\widehat{H}^{-3}(G_{i},\mathbbm{Z})\xrightarrow{\rm
cores}\widehat{H}^{-3}(\overline{G}_{1},\mathbbm{Z})$ is surjective, it
follows from Theorem 6.5 that ${\rm Obs}(K/k)={\rm
Obs}_{1}(\overline{L}_{1}/K/k)$. We also checked that ${\rm
Ker}\,\overline{\psi}_{1}/\overline{\varphi}_{1}({\rm
Ker}\,\overline{\psi}_{1}^{\rm nr})$ $\simeq$ $\mathbbm{Z}/2\mathbbm{Z}$ for
$\overline{G}_{1}$ and ${\rm
Ker}\,\overline{\psi}_{1}/\overline{\varphi}_{1}({\rm
Ker}\,\overline{\psi}_{2}^{\rm nr})=0$ for $\overline{G}_{j}$ $(j\in J)$. This
implies that ${\rm Obs}(K/k)\neq{\rm Obs}_{1}(\overline{L}_{j}/K/k)$ when
$\overline{L}_{j}/k$ is unramified for $j\in J$. Apply
FirstObstructionDr($\overline{G}_{1},\overline{G}_{1}^{\prime},\overline{H}_{1}$)
for all subgroups $\overline{G}_{1}^{\prime}\leq\overline{G}_{1}$. We find
that ${\rm Obs}_{1}(\overline{L}_{1}/K/k)$ $=$ $1$ if and only if there exists
$v\in V_{k}$ such that (i) $V_{4}\leq G_{v}$ where $|{\rm Orb}_{V_{4}}(i)|=2$
for any $1\leq i\leq 8$ and $V_{4}\cap[G,G]=1$ $($equivalently, $|{\rm
Orb}_{V_{4}}(i)|=2$ for any $1\leq i\leq 8$ and $V_{4}$ is not in $A_{8}$$)$
or (ii) $C_{4}\times C_{2}\leq G_{v}$ where $(C_{4}\times
C_{2})\cap[G,G]\simeq C_{2}$ $($equivalently, $C_{4}\times C_{2}$ is
transitive in $S_{8}$$)$ (see Details 6.8 and Example 6.12).

(2-4) The case $G=8T19\simeq(C_{2})^{3}\rtimes C_{4}$. We have $H\simeq C_{4}$
and $M(G)\simeq(\mathbbm{Z}/2\mathbbm{Z})^{\oplus 2}$. Applying the function
MinimalStemExtensions($G$)[$j$].MinimalStemExtension, we get the minimal stem
extensions $\overline{G}_{1},\overline{G}_{2},\overline{G}_{3}$ of $G$. Use
Theorem 6.5. Applying KerResH3Z(G,H), we see that ${\rm
Ker}\\{H^{3}(\overline{G}_{3},\mathbbm{Z})\xrightarrow{\rm
res}\oplus_{i=1}^{m^{\prime}}H^{3}(G_{i},\mathbbm{Z})\\}=0$ but ${\rm
Ker}\\{H^{3}(\overline{G}_{j},\mathbbm{Z})\xrightarrow{\rm
res}\oplus_{i=1}^{m^{\prime}}H^{3}(G_{i},\mathbbm{Z})\\}=\mathbbm{Z}/2\mathbbm{Z}$
for $j\in J:=\\{1,2\\}$. Because
$\oplus_{i=1}^{m^{\prime}}\widehat{H}^{-3}(G_{i},\mathbbm{Z})\xrightarrow{\rm
cores}\widehat{H}^{-3}(\overline{G}_{3},\mathbbm{Z})$ is surjective, it
follows from Theorem 6.5 that ${\rm Obs}(K/k)={\rm
Obs}_{1}(\overline{L}_{3}/K/k)$. We also checked that ${\rm
Ker}\,\overline{\psi}_{1}/\overline{\varphi}_{1}({\rm
Ker}\,\overline{\psi}_{1}^{\rm nr})$ $\simeq$ $\mathbbm{Z}/2\mathbbm{Z}$ for
$\overline{G}_{3}$ and ${\rm
Ker}\,\overline{\psi}_{1}/\overline{\varphi}_{1}({\rm
Ker}\,\overline{\psi}_{2}^{\rm nr})=0$ for $\overline{G}_{j}$ $(j\in J)$. This
implies that ${\rm Obs}(K/k)\neq{\rm Obs}_{1}(\overline{L}_{j}/K/k)$ when
$\overline{L}_{j}/k$ is unramified for $j\in J$. Apply
FirstObstructionDr($\overline{G}_{3},\overline{G}_{3}^{\prime},\overline{H}_{3}$)
for all subgroups $\overline{G}_{3}^{\prime}\leq\overline{G}_{3}$. We find
that ${\rm Obs}_{1}(\overline{L}_{3}/K/k)$ $=$ $1$ if and only if there exists
$v\in V_{k}$ such that (i) $V_{4}\leq G_{v}$ where $V_{4}\cap Z(G)=1$ and
$V_{4}\cap Z^{2}(G)\simeq C_{2}$ with the upper central series $1\leq Z(G)\leq
Z^{2}(G)\leq G$ of $G$ $($equivalently, $|{\rm Orb}_{V_{4}}(i)|=4$ for any
$1\leq i\leq 8$ and $V_{4}\cap Z(G)=1$$)$; or (ii) $C_{4}\times C_{2}\leq
G_{v}$ where $C_{4}\times C_{2}$ is not transitive in $S_{8}$ or $[G,G]\leq
C_{4}\times C_{2}$ (see Details 6.8 and Example 6.12).

(2-5) The case $G=8T22\simeq(C_{2})^{3}\rtimes V_{4}$. We have $H\simeq V_{4}$
and $M(G)\simeq(\mathbbm{Z}/2\mathbbm{Z})^{\oplus 5}$. Applying the function
MinimalStemExtensions($G$)[$j$].MinimalStemExtension, we get the minimal stem
extensions $\overline{G}_{1},\ldots,\overline{G}_{31}$ of $G$. Use Theorem
6.5. Applying KerResH3Z(G,H), we see that ${\rm
Ker}\\{H^{3}(\overline{G}_{16},\mathbbm{Z})\xrightarrow{\rm
res}\oplus_{i=1}^{m^{\prime}}H^{3}(G_{i},\mathbbm{Z})\\}=0$ but ${\rm
Ker}\\{H^{3}(\overline{G}_{j},\mathbbm{Z})\xrightarrow{\rm
res}\oplus_{i=1}^{m^{\prime}}H^{3}(G_{i},\mathbbm{Z})\\}\neq 0$ for $j\in
J:=\\{j\mid 1\leq j\leq 31,j\neq 16\\}$. Because
$\oplus_{i=1}^{m^{\prime}}\widehat{H}^{-3}(G_{i},\mathbbm{Z})\xrightarrow{\rm
cores}\widehat{H}^{-3}(\overline{G}_{16},\mathbbm{Z})$ is surjective, it
follows from Theorem 6.5 that ${\rm Obs}(K/k)={\rm
Obs}_{1}(\overline{L}_{16}/K/k)$. We also checked that ${\rm
Ker}\,\overline{\psi}_{1}/\overline{\varphi}_{1}({\rm
Ker}\,\overline{\psi}_{1}^{\rm nr})$ $\simeq$ $\mathbbm{Z}/2\mathbbm{Z}$ for
$\overline{G}_{16}$ and ${\rm
Ker}\,\overline{\psi}_{1}/\overline{\varphi}_{1}({\rm
Ker}\,\overline{\psi}_{2}^{\rm nr})=0$ for $\overline{G}_{j}$ $(j\in J)$. This
implies that ${\rm Obs}(K/k)\neq{\rm Obs}_{1}(\overline{L}_{j}/K/k)$ when
$\overline{L}_{j}/k$ is unramified for $j\in J$. Apply
FirstObstructionDr($\overline{G}_{16},\overline{G}_{16}^{\prime},\overline{H}_{16}$)
for all subgroups $\overline{G}_{16}^{\prime}\leq\overline{G}_{16}$. We find
that ${\rm Obs}_{1}(\overline{L}_{16}/K/k)$ $=$ $1$ if and only if there
exists $v\in V_{k}$ such that (i) $V_{4}\leq G_{v}$ where $|{\rm
Orb}_{V_{4}}(i)|=4$ for any $1\leq i\leq 8$ and $V_{4}\cap Z(G)=1$ or (ii)
$C_{4}\times C_{2}\leq G_{v}$ where $C_{4}\times C_{2}$ is transitive in
$S_{8}$. (see Details 6.8 and Example 6.12).

(2-6) The case $G=8T32\simeq((C_{2})^{3}\rtimes V_{4})\rtimes C_{3}$. We have
$H\simeq A_{4}$ and $M(G)\simeq(\mathbbm{Z}/2\mathbbm{Z})^{\oplus 3}$.
Applying the function MinimalStemExtensions($G$)[$j$].MinimalStemExtension, we
get the minimal stem extensions $\overline{G}_{1},\ldots,\overline{G}_{7}$ of
$G$. Use Theorem 6.5. Applying KerResH3Z(G,H), we see that ${\rm
Ker}\\{H^{3}(\overline{G}_{5},\mathbbm{Z})\xrightarrow{\rm
res}\oplus_{i=1}^{m^{\prime}}H^{3}(G_{i},\mathbbm{Z})\\}=0$ but ${\rm
Ker}\\{H^{3}(\overline{G}_{j},\mathbbm{Z})\xrightarrow{\rm
res}\oplus_{i=1}^{m^{\prime}}H^{3}(G_{i},\mathbbm{Z})\\}\simeq\mathbbm{Z}/2\mathbbm{Z}$
for $j\in J:=\\{j\mid 2\leq j\leq 7\\}$. Because
$\oplus_{i=1}^{m^{\prime}}\widehat{H}^{-3}(G_{i},\mathbbm{Z})\xrightarrow{\rm
cores}\widehat{H}^{-3}(\overline{G}_{5},\mathbbm{Z})$ is surjective, it
follows from Theorem 6.5 that ${\rm Obs}(K/k)={\rm
Obs}_{1}(\overline{L}_{5}/K/k)$. We also checked that ${\rm
Ker}\,\overline{\psi}_{1}/\overline{\varphi}_{1}({\rm
Ker}\,\overline{\psi}_{1}^{\rm nr})$ $\simeq$ $\mathbbm{Z}/2\mathbbm{Z}$ for
$\overline{G}_{5}$ and ${\rm
Ker}\,\overline{\psi}_{1}/\overline{\varphi}_{1}({\rm
Ker}\,\overline{\psi}_{2}^{\rm nr})=0$ for $\overline{G}_{j}$ $(j\in J)$. This
implies that ${\rm Obs}(K/k)\neq{\rm Obs}_{1}(\overline{L}_{j}/K/k)$ when
$\overline{L}_{j}/k$ is unramified for $j\in J$. Apply
FirstObstructionDr($\overline{G}_{1},\overline{G}_{1}^{\prime},\overline{H}_{1}$)
for all subgroups $\overline{G}_{1}^{\prime}\leq\overline{G}_{1}$. We find
that ${\rm Obs}_{1}(\overline{L}_{1}/K/k)$ $=$ $1$ if and only if there exists
$v\in V_{k}$ such that (i) $V_{4}\leq G_{v}$ where $|{\rm Orb}_{V_{4}}(i)|=4$
for any $1\leq i\leq 8$ and $V_{4}\cap Z(G)=1$ or (ii) $C_{4}\times C_{2}\leq
G_{v}$ where $C_{4}\times C_{2}$ is transitive in $S_{8}$ (see Details 6.8 and
Example 6.12).

(3) $n=9$: $G=9Tm$ $(m=2,5,7,9,11,14,23)$. Applying FirstObstructionN($G$), we