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# Chain-imprimitive, flag-transitive $2$-designs

Carmen Amarraa,b∗ ,  Alice Devillersa and  Cheryl E. Praegera aCentre for the
Mathematics of Symmetry and Computation, The University of Western Australia,
35 Stirling Highway, Crawley, Western Australia 6009, Australia
<EMAIL_ADDRESS><EMAIL_ADDRESS>b _Permanent address_ :
Institute of Mathematics, University of the Philippines Diliman, C.P. Garcia
Avenue, Quezon City 1101, Philippines<EMAIL_ADDRESS>∗Corresponding
author

###### Abstract.

We consider $2$-designs which admit a group of automorphisms that is flag-
transitive and leaves invariant a chain of nontrivial point-partitions. We
build on our recent work on $2$-designs which are block-transitive but not
necessarily flag-transitive. In particular we use the concept of the “array”
of a point subset with respect to the chain of point-partitions; the array
describes the distribution of the points in the subset among the classes of
each partition. We obtain necessary and sufficient conditions on the array in
order for the subset to be a block of such a design. By explicit construction
we show that for any $s\geq 2$, there are infinitely many $2$-designs
admitting a flag-transitive group that preserves an invariant chain of point-
partitions of length $s$. Moreover an exhaustive computer search, using Magma,
seeking designs with $e_{1}e_{2}e_{3}$ points (where each $e_{i}\leq 50$) and
a partition chain of length $s=3$, produced $57$ such flag-transitive designs,
among which only three designs arise from our construction – so there is still
much to learn.

##  1\. Introduction

We investigate $2$-designs which admit an automorphism group that preserves a
chain structure on its point set. Thus we are interested in
$2$-$(v,k,\lambda)$ designs ${\mathscr{D}}$, namely incidence structures that
consist of a _point set_ ${\mathscr{P}}$ and a _block set_ ${\mathscr{B}}$ of
subsets of ${\mathscr{P}}$ with the property that $|{\mathscr{P}}|=v$, each
block has size $k$, and any two distinct points are contained in exactly
$\lambda$ blocks. A subgroup $G\leq{\mathrm{Aut}}({\mathscr{D}})$ is said to
act _$s$ -chain-imprimitively_ on ${\mathscr{P}}$ if $G$ is transitive on
${\mathscr{P}}$ and leaves invariant each partition in an $s$-chain

(1) $\bm{\mathscr{C}}:\
{\textstyle\binom{{\mathscr{P}}}{1}}={\mathscr{C}}_{0}\prec{\mathscr{C}}_{1}\prec\ldots\prec{\mathscr{C}}_{s-1}\prec{\mathscr{C}}_{s}=\\{{\mathscr{P}}\\}$

of partitions of ${\mathscr{P}}$. (Here $\binom{{\mathscr{P}}}{1}$ denotes the
partition of ${\mathscr{P}}$ into singletons, and
${\mathscr{C}}_{i-1}\prec{\mathscr{C}}_{i}$ means that each
${\mathscr{C}}_{i-1}$-class is a proper subset of some
${\mathscr{C}}_{i}$-class.) Any point-imprimitive design is $2$-chain-
imprimitive and recent results in [2], that both characterised parameters and
gave explicit examples, showed that _there exist $G$-block-transitive
$2$-$(v,k,\lambda)$ designs ${\mathscr{D}}=({\mathscr{P}},{\mathscr{B}})$ with
$G$ acting $s$-chain-imprimitively on ${\mathscr{P}}$ for which the chain
length $s$ is arbitrarily large_. A question arose from the work in [2] as to
whether the same assertion would hold true if we required also that $G$ should
be flag-transitive, not only block-transitive. Confirming this assertion is
the aim of this paper.

A _flag_ of a $2$-$(v,k,\lambda)$ design
${\mathscr{D}}=({\mathscr{P}},{\mathscr{B}})$ is a pair
$(\delta,B)\in{\mathscr{P}}\times{\mathscr{B}}$ such that $\delta\in B$, and
${\mathscr{D}}$ is _$G$ -flag-transitive_ (for
$G\leq{\mathrm{Aut}}({\mathscr{D}})$) if $G$ acts transitively on the set of
flags. Flag-transitivity of $G$ implies that $G$ is both block-transitive and
point-transitive, but the converse does not hold in general. Moreover, Davis
[9] showed that for each value of the parameter $\lambda$ there are only
finitely many possible ${\mathscr{D}}$ with $G$ point-imprimitive and flag-
transitive. So we were not at all sure when we began this study that we would
find examples of $G$-flag-transitive designs with $G$ acting $s$-chain-
imprimitively on ${\mathscr{P}}$ and the chain length $s$ unboundedly large.
None of the block-transitive designs constructed in [2] were flag-transitive,
and although there are many papers in the literature contributing to the
classification of flag-transitive, point-imprimitive $2$-designs, for example
[5, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21], we did not find any
constructions in them which would help to answer this question. Building on
the theory developed in [2] we were able to prove the following result.

######  Theorem 1.1.

For any integer $s\geq 2$, there exist infinitely many $2$-designs
${\mathscr{D}}=({\mathscr{P}},{\mathscr{B}})$ such that some subgroup
$G\leq{\mathrm{Aut}}({\mathscr{D}})$ acts flag-transitively on ${\mathscr{D}}$
and $s$-chain-imprimitively on ${\mathscr{P}}$.

In order to prove this theorem, which ultimately relies on an explicit
construction (see Constructions 4.2 and 6.1), we needed to develop further the
theoretical framework introduced in [2]. Suppose that
${\mathscr{D}}=({\mathscr{P}},{\mathscr{B}})$ is a $2$-$(v,k,\lambda)$ design
and $G\leq{\mathrm{Aut}}({\mathscr{D}})$ is flag-transitive on ${\mathscr{D}}$
and $s$-chain-imprimitive on ${\mathscr{P}}$, preserving the partitions in an
$s$-chain $\bm{\mathscr{C}}$ as in (1). Parameters of significance are the
positive integers $e_{1},\dots,e_{s}$ such that, for $1\leq i\leq s$, each
class of ${\mathscr{C}}_{i}$ contains $e_{i}$ classes of
${\mathscr{C}}_{i-1}$. Then $v=e_{1}\dots e_{s}$, and in the next result
Theorem 1.2 we identify necessary and sufficient conditions on the parameters
$v,k,s,e_{1},\dots,e_{s}$ for the existence of a flag-transitive, $s$-chain-
imprimitive $2$-design.

######  Theorem 1.2.

Let $s,k,e_{1},\ldots,e_{s}$ be integers, all at least $2$, such that $k<v$,
where $v:=\prod_{i=1}^{s}e_{i}$. Let ${\mathscr{P}}$ be a set of size $v$ and
$\bm{\mathscr{C}}$ a chain of partitions as in (1), such that for $1\leq i\leq
s$ each class of ${\mathscr{C}}_{i}$ contains $e_{i}$ classes of
${\mathscr{C}}_{i-1}$. Let $d:=\gcd(e_{1}-1,\ldots,e_{s}-1)$, and for each
$i\in\\{1,\ldots,s\\}$ let

(2)
$y_{i}:=1+\frac{k-1}{v-1}\left(\left(\prod_{j=1}^{i}e_{j}\right)-1\right)\quad\mbox{(so
in particular $y_{s}=k$)}.$

Then there exists a $2$-$(v,k,\lambda)$ design ${\mathscr{D}}$ with point set
${\mathscr{P}}$, for some $\lambda$, and a group
$G\leq{\mathrm{Aut}}({\mathscr{D}})$ such that $G$ is flag-transitive on
${\mathscr{D}}$ and preserves the partitions in the $s$-chain
$\bm{\mathscr{C}}$, if and only if the following conditions both hold:

  * (FT1)

$v-1$ divides $(k-1)\cdot d$; and

  * (FT2)

for each $i\in\\{1,\ldots,s-1\\}$, $y_{i}$ is a positive integer dividing
$(e_{i+1}-1)\left(\prod_{j=1}^{i}e_{j}\right)/d$.

Moreover if these two conditions hold and
${\mathscr{D}}=({\mathscr{P}},{\mathscr{B}})$ is such a design then, for each
$i\in\\{1,\ldots,s\\}$, each class $C\in{\mathscr{C}}_{i}$ and each block
$B\in{\mathscr{B}}$, the intersection size $|B\cap C|\in\\{0,y_{i}\\}$.

In Theorem 1.2, the flag-transitive group $G$ is contained in the full
stabiliser of the partition chain $\bm{\mathscr{C}}$, namely the iterated
wreath product $W=S_{e_{1}}\wr\ldots\wr S_{e_{s}}$. Thus (see, for example,
[6, Proposition 1.1]), if we replace $G$ by its over-group $W$, and replace
the block set ${\mathscr{B}}$ (which is the $G$-orbit $B^{G}=\\{B^{g}\mid g\in
G\\}$ of a subset $B\subseteq{\mathscr{P}}$) by the possibly larger $W$-orbit
$B^{W}$, then ${\mathscr{D}}^{\prime}=({\mathscr{P}},B^{W})$ is also a
$2$-$(v,k,\lambda^{\prime})$ design with
$W\leq{\mathrm{Aut}}({\mathscr{D}}^{\prime})$ flag-transitive and preserving
the same $s$-chain $\bm{\mathscr{C}}$ of partitions of ${\mathscr{P}}$. So
${\mathscr{D}}^{\prime}$ will have the same symmetry properties, the same
point set ${\mathscr{P}}$ and partition chain $\bm{\mathscr{C}}$, the same
parameters $v,k,s,e_{1},\dots,e_{s}$, and possibly larger
$\lambda^{\prime}\geq\lambda$. Thus, in order to identify in Theorem 1.2
necessary and sufficient conditions on the parameters
$v,k,s,e_{1},\dots,e_{s}$ for the existence of a flag-transitive, $s$-chain-
imprimitive $2$-design, we may work with the larger group $W$ and design
${\mathscr{D}}^{\prime}$. Similarly, in order to prove Theorem 1.1, we find,
for each $s$, infinitely many parameter sequences $k,e_{1},\dots,e_{s}$
satisfying the conditions (i) and (ii) of Theorem 1.2. Moreover, we explicitly
exhibit a $k$-subset $B$ satisfying the conditions in the final sentence of
Theorem 1.2, and prove that $({\mathscr{P}},B^{W})$ is a $2$-design with the
required properties (see Constructions 4.2 and 6.1; and Propositions 4.3 and
6.2).

We end this section with some comments in Subsection 1.1 about the design
constructions and some open questions. The organisation of the rest of paper
is as follows: Section 2 contains necessary background work about iterated
wreath products $W$ and their action on ${\mathscr{P}}$. Section 3 contains
some specific results about ‘uniform’ point-sets, which are used in Section 4
to justify our general construction method for flag-transitive $s$-chain-
imprimitive $2$-designs given in Construction 4.2. In Section 5 we prove
Theorem 1.2, and in the final Section 6 we present explicit families of
examples, and in particular we prove Theorem 1.1.

###  1.1. Comments on our constructions and results

We make several comments about constructions of flag-transitive $s$-chain-
imprimitive $2$-designs, and in particular relate our work to earlier results.

(a) _Comment on Theorem 1.2 for the case $s=2$._ Theorem 1.2 for the case
$s=2$ can be derived from several results in [6] as follows. If $s=2$ then the
number of points is $v=|{\mathscr{P}}|=e_{1}e_{2}$. The chain
$\bm{\mathscr{C}}$ in (1) has just one nontrivial partition, namely
${\mathscr{C}}_{1}=\\{C_{1},\dots,C_{e_{2}}\\}$, with $e_{2}$ classes of size
$e_{1}$. The point-block incidence structure
${\mathscr{D}}={\mathscr{D}}(e_{1},e_{2};\mathbf{x})=({\mathscr{P}},{\mathscr{B}})$
defined in [6, Section 2], where $\mathbf{x}=(x_{1},\dots,x_{e_{2}})$ with
non-negative integer entries $x_{1}\geq\ldots\geq x_{e_{2}}$, has as block set
${\mathscr{B}}$ the collection of all subsets $B\subseteq{\mathscr{P}}$ such
that the sequence $\big{(}\,|B\cap C_{i}|\ |\ 1\leq i\leq e_{2}\,\big{)}$ is a
permutation of the sequence $\mathbf{x}$. Thus $k:=|B|=\sum_{i}x_{i}$, and
${\mathrm{Aut}}({\mathscr{D}})$ is the wreath product $S_{e_{1}}\wr
S_{e_{2}}$. As noted at the beginning of the proof of [6, Proposition 4.1], a
necessary and sufficient condition for ${\mathscr{D}}(e_{1},e_{2};\mathbf{x})$
to be flag-transitive is that the tuple $\mathbf{x}$ is some rearrangement of
$(\ell^{k/\ell},0^{e_{2}-k/\ell})$, for a divisor $\ell$ of $k$ with
$1<\ell<k$. Then the criterion given by [6, Proposition 2.2(ii)] for
${\mathscr{D}}$ to be a $2$-design is that $k(\ell-1)=k(k-1)(e_{1}-1)/(v-1)$,
that is $\ell=1+(k-1)(e_{1}-1)/(v-1)$. Thus $\ell$ is the parameter $y_{1}$ in
(2). Since $\ell$ is a positive integer this equation implies that $v-1$
divides $(k-1)(e_{1}-1)$. Note that
$\gcd(v-1,e_{1}-1)=\gcd(e_{1}e_{2}-1,e_{1}-1)=\gcd(e_{2}-1,e_{1}-1)$, which is
the parameter $d$ of Theorem 1.2. Hence $v-1$ divides $(k-1)d$ which is
condition (i) of Theorem 1.2. To obtain condition (ii), we first subtract the
equation $\ell=1+(k-1)(e_{1}-1)/(v-1)$ from the equation
$k=1+(k-1)(e_{1}e_{2}-1)/(v-1)$ to obtain $k-\ell=z$, where
$z=(k-1)(e_{2}-1)e_{1}/(v-1)$. Note that $z$ is a positive integer, and as
$\ell$ divides $k$, $\ell$ must also divide $z$. Also
$\gcd(e_{2}-1,v-1)=\gcd(e_{2}-1,e_{1}-1)=d$ and $v-1$ is coprime to $e_{1}$,
so $z$ factorises as

$z=\frac{(e_{2}-1)e_{1}}{d}\cdot\frac{k-1}{(v-1)/d}.$

Then, again as $\ell$ divides $k$, $\ell$ is coprime to the second factor and
hence $\ell$ divides $(e_{2}-1)e_{1}/d$, which is condition (ii). Thus
existence of a flag-transitive point-imprimitive $2$-design implies conditions
(i) and (ii), and reversing this argument we see that if conditions (i) and
(ii) hold then the design ${\mathscr{D}}(e_{1},e_{2};\mathbf{x})$, with
$\mathbf{x}=(\ell^{k/\ell},0^{e_{2}-k/\ell})$ where
$\ell=1+(k-1)(e_{1}-1)/(v-1)$, gives an example of a $2$-design with the
required properties.

(b) _Comment on Theorem 1.1 for the case $s\geq 3$._ For each $s\geq 3$ we can
obtain a flag-transitive, $(s-1)$-chain-imprimitive $2$-design from a flag-
transitive, $s$-chain-imprimitive $2$-design, as follows: Given a point set
${\mathscr{P}}$ with an $s$-chain $\bm{\mathscr{C}}$ of partitions
${\mathscr{C}}_{1},\ldots,{\mathscr{C}}_{s}$, with parameters
$e_{1},\ldots,e_{s}$ as in Theorem 1.2, we form a new chain
$\bm{\mathscr{C}}^{\prime}$ of partitions
${\mathscr{C}}^{\prime}_{0},\ldots,{\mathscr{C}}^{\prime}_{s-1}$ of
${\mathscr{P}}$ by deleting any nontrivial partition ${\mathscr{C}}_{i}$
($1\leq i\leq s-1$) from $\bm{\mathscr{C}}$, so that
${\mathscr{C}}^{\prime}_{j}={\mathscr{C}}_{j}$ for $0\leq j\leq i-1$ and
${\mathscr{C}}^{\prime}_{j}={\mathscr{C}}_{j+1}$ for $i\leq j\leq s-1$. Let
$e^{\prime}_{j}:=e_{j}$ for $0\leq j\leq i-2$,
$e^{\prime}_{i-1}:=e_{i}e_{i-1}$, and $e^{\prime}_{j}:=e_{j+1}$ for $i\leq
j\leq s-1$. Then the partition ${\mathscr{C}}^{\prime}_{j}$ contains
$e^{\prime}_{j}$ classes of ${\mathscr{C}}^{\prime}_{j-1}$, for $1\leq j\leq
s-1$. Suppose that ${\mathscr{D}}=\big{(}{\mathscr{P}},B^{G}\big{)}$ is a
$G$-flag-transitive $2$-design such that $G$ is $s$-chain-imprimitive on
${\mathscr{P}}$, preserving the $s$-chain $\bm{\mathscr{C}}$, where $B$ is a
$k$-subset of ${\mathscr{P}}$. Then $G\leq W:=S_{e_{1}}\wr\ldots\wr
S_{e_{s}}$, the stabiliser of the $s$-chain $\bm{\mathscr{C}}$. We first
replace ${\mathscr{B}}=B^{G}$ by the possibly larger block set $B^{W}$ as
explained in the paragraph after the statement of Theorem 1.2, and note that
$W$ leaves the $(s-1)$-chain $\bm{\mathscr{C}}^{\prime}$ invariant, so $W$ is
a subgroup of the stabiliser $H:=S_{e^{\prime}_{1}}\wr\ldots\wr
S_{e^{\prime}_{s-1}}$ of $\bm{\mathscr{C}}^{\prime}$. Define
${\mathscr{D}}^{\prime}:=\big{(}{\mathscr{P}},B^{H}\big{)}$. Then
${\mathscr{D}}^{\prime}$ is an $H$-flag-transitive $2$-design (see, for
example, [6, Proposition 1.1]), and by definition, $H$ is $(s-1)$-chain-
imprimitive on ${\mathscr{P}}$, preserving the $(s-1)$-chain
$\bm{\mathscr{C}}^{\prime}$.

(c) _Comment on further examples._ Using Theorem 1.2 and Magma [4], we were
able to find all parameter sets $(e_{1},e_{2},e_{3};k)$ with
$e_{1},e_{2},e_{3}\leq 50$ which correspond to flag-transitive, $3$-chain-
imprimitive $2$-designs with automorphism group $G=S_{e_{1}}\wr S_{e_{2}}\wr
S_{e_{3}}$ and block size $k$. These are listed in Table 1; the entries in
boldface in Table 1 correspond to parameters of designs obtained using
Construction 6.1. We see from Table 1 that there are many more explicit
feasible parameter sets apart from those used in Construction 6.1, and this
suggests that there may be other methods for construction of infinite families
of flag-transitive, $s$-chain-imprimitive $2$-designs.

######  Problem 1.

Find more infinite families of flag-transitive, $s$-chain-imprimitive
$2$-designs, with $s\geq 3$. In particular, find more families with $s$
unbounded.

_Data availability statement._ No datasets were generated or analysed in the
current study.

_Conflicts of interest._ All authors have no conflicts of interest to
disclose.

##  2\. Chain imprimitivity

###  2.1. Chain structure on a set

We shall use the same notation and terminology as in [2] for the labelling of
the point set ${\mathscr{P}}$ and the partition chain $\bm{\mathscr{C}}$.

Given integers $s\geq 2$ and $e_{1},\ldots,e_{s}\geq 2$. The set
${\mathscr{P}}$ is the Cartesian product $\prod_{i=1}^{s}\mathbb{Z}_{e_{i}}$.

For any $i\in\\{0,\ldots,s\\}$ and any
$(\delta_{j})_{j>i}\in\prod_{j>i}\mathbb{Z}_{e_{j}}$, we define the subset
$C_{(\delta_{j})_{j>i}}$ of ${\mathscr{P}}$ as follows:

(3)
$C_{(\delta_{j})_{j>i}}:=\big{\\{}(\varepsilon_{j})_{j=1}^{s}\in{\mathscr{P}}\
\big{|}\ \varepsilon_{j}=\delta_{j}\ \text{for all}\ j>i\big{\\}}.$

When $i=0$, $(\delta_{j})_{j>0}=(\delta_{1},\delta_{2},\ldots,\delta_{s})$ is
a point of ${\mathscr{P}}$ and $C_{(\delta_{j})_{j>i}}$ is the singleton set
containing that point. When $i=s$, $(\delta_{j})_{j>s}=(\,)$ is the unique
empty tuple, and the membership condition (3) for
$C_{(\delta_{j})_{j>s}}=C_{(\,)}$ is vacuous, and hence
$C_{(\delta_{j})_{j>s}}={\mathscr{P}}$. By convention
$\prod_{j>s}\mathbb{Z}_{e_{j}}$ is the singleton set $\\{(\,)\\}$. For any
$i\in\\{0,\ldots,s\\}$, we then define the family ${\mathscr{C}}_{i}$ of
subsets by

(4) ${\mathscr{C}}_{i}:=\Big{\\{}C_{(\delta_{j})_{j>i}}\ \Big{|}\
(\delta_{j})_{j>i}\in\prod_{j>i}\mathbb{Z}_{e_{j}}\Big{\\}}.$

Then ${\mathscr{C}}_{0}$ and ${\mathscr{C}}_{s}$ are trivial partitions of
${\mathscr{P}}$ consisting of the singleton subsets of ${\mathscr{P}}$ and of
the entire set ${\mathscr{P}}$, respectively, while
${\mathscr{C}}_{1},\ldots,{\mathscr{C}}_{s-1}$ are nontrivial partitions of
${\mathscr{P}}$. For each $i\in\\{0,\ldots,s-1\\}$ the partition
${\mathscr{C}}_{i}$ is a proper refinement of ${\mathscr{C}}_{i+1}$, which we
write as ${\mathscr{C}}_{i}\prec{\mathscr{C}}_{i+1}$. Thus, under the partial
order $\prec$, the partitions ${\mathscr{C}}_{0},\ldots,{\mathscr{C}}_{s}$
form a chain $\bm{\mathscr{C}}$ as in (1). Moreover each set ${\mathscr{P}}$
and chain $\bm{\mathscr{C}}$ with parameters $e_{1},\dots,e_{s}$ can be
labelled in this way.

Each partition ${\mathscr{C}}_{i}$ has parameters $c_{i}$, $d_{i}$, and
$e_{i}$, where $c_{i}$ is the size of each class in ${\mathscr{C}}_{i}$,
$d_{i}$ is the number of classes in ${\mathscr{C}}_{i}$, $e_{0}:=1$, and, for
each $i\in\\{1,\ldots,s\\}$, $e_{i}$ is the number of
${\mathscr{C}}_{i-1}$-classes contained in each ${\mathscr{C}}_{i}$-class.
Thus

$c_{i}=\prod_{j=0}^{i}e_{j}\quad\text{and}\quad d_{i}=\prod_{j=i+1}^{s}e_{j}.$

In particular $c_{0}=d_{s}=1$, $c_{s}=d_{0}=|{\mathscr{P}}|$, $c_{1}=e_{1}$,
and $d_{s-1}=e_{s}$.

For $i\in\\{0,\ldots,s\\}$ and any nonempty subset $X\subseteq{\mathscr{P}}$,
we define

(5) ${\mathscr{C}}_{i}(X):=\\{C\in{\mathscr{C}}_{i}\ |\ X\cap
C\neq\varnothing\\}.$

In particular, if $C\in{\mathscr{C}}_{i}$, then ${\mathscr{C}}_{i-1}(C)$ is
the set of all ${\mathscr{C}}_{i-1}$-classes that are contained in $C$, and
${\mathscr{C}}_{i+1}(C)=\\{C^{+}\\}$, where

$C^{+}:=\text{the unique ${\mathscr{C}}_{i+1}$-class that contains
$C\in{\mathscr{C}}_{i}$}.$

It follows that for any subset $X\subseteq{\mathscr{P}}$ and any class
$C\in{\mathscr{C}}_{i}$, the set of all ${\mathscr{C}}_{i-1}$-classes that are
contained in $C$ and that contain points of $X$ is

$\displaystyle{\mathscr{C}}_{i-1}(X)\cap{\mathscr{C}}_{i-1}(C)$
$\displaystyle=\\{C^{\prime}\in{\mathscr{C}}_{i-1}\ |\ X\cap
C^{\prime}\neq\varnothing\ \text{and}\ C^{\prime}\subseteq C\\}$
$\displaystyle=\\{C^{\prime}\in{\mathscr{C}}_{i-1}\ |\ C^{\prime}\cap X\cap
C\neq\varnothing\\}$ $\displaystyle={\mathscr{C}}_{i-1}(X\cap C).$

###  2.2. Iterated wreath product of groups

Using the notation of Subsection 2.1, for each $i\in\\{1,\ldots,s\\}$ let
$G_{i}\leq{\mathrm{Sym}}(\mathbb{Z}_{e_{i}})=S_{e_{i}}$. We denote by
$G:=G_{1}\wr\ldots\wr G_{s}$ the _iterated wreath product_ of the groups
$G_{1},\ldots,G_{s}$. For details of the definition of the group $G$, its
binary operation, and its actions on ${\mathscr{P}}$ and on each of the
partitions ${\mathscr{C}}_{1},\ldots,{\mathscr{C}}_{s}$ we refer the reader to
the treatment in [2, Subsection 2.3], based on [3].

The following fact will be useful in our proofs. Observe that for $s\geq 2$,
$1\leq j\leq s-1$, and for each $C\in{\mathscr{C}}_{j}$,

(6) $\bm{\mathscr{C}}(C):\
{\textstyle\binom{C}{1}}={\mathscr{C}}_{0}(C)\prec{\mathscr{C}}_{1}(C)\prec\ldots\prec{\mathscr{C}}_{j-1}(C)\prec{\mathscr{C}}_{j}(C)=\\{C\\}$

is a $j$-chain on $C$. For a group $H$ acting on a set $X$ we denote by
$H^{X}$ the subgroup of ${\mathrm{Sym}}(X)$ induced by $H$.

######  Proposition 2.1.

Let $s\geq 2$, $1\leq i\leq s$, and $C\in{\mathscr{C}}_{i}$, and let
$G=S_{e_{1}}\wr\ldots\wr S_{e_{s}}$.

  1. (a)

The stabiliser in ${\mathrm{Sym}}(C)$ of the chain $\bm{\mathscr{C}}(C)$ in
(6) is $H(C):=G_{C}^{C}\cong S_{e_{1}}\wr\ldots\wr S_{e_{i}}$, and
$H(C)^{{\mathscr{C}}_{i-1}(C)}\cong S_{e_{i}}$.

  2. (b)

Further, for $1\leq j\leq i-1$,
$H(C)\cong\left(\prod_{C^{\prime}\in{\mathscr{C}}_{j}(C)}H(C^{\prime})\right)\rtimes
H(C)^{{\mathscr{C}}_{j}(C)}$, where $H(C)^{{\mathscr{C}}_{j}(C)}\cong
S_{e_{j+1}}\wr\ldots\wr S_{e_{i}}$ and
$|{\mathscr{C}}_{j-1}(C)|=\prod_{\ell=j}^{i}e_{\ell}$.

  3. (c)

In particular, for $1\leq j\leq s-1$,
$G=\left(\prod_{C^{\prime}\in{\mathscr{C}}_{j}}H(C^{\prime})\right)\rtimes
G^{{\mathscr{C}}_{j}}$, where $G^{{\mathscr{C}}_{j}}\cong
S_{e_{j+1}}\wr\ldots\wr S_{e_{s}}$ and
$|{\mathscr{C}}_{j}|=d_{j}=\prod_{i=j+1}^{s}e_{i}$.

###### Proof.

(a) This follows from [2, Theorem 2.1 (i)] applied to ${\mathscr{C}}(C)$, and
from [2, Theorem 2.1 (ii)] applied to $H(C)^{{\mathscr{C}}_{j-1}(C)}$.

(b) By the discussion before [2, Theorem 2.1] applied to $H(C)$, the kernel
$H(C)_{({\mathscr{C}}_{j}(C))}$ of the $H(C)$-action on ${\mathscr{C}}_{j}(C)$
is
$H(C)_{({\mathscr{C}}_{j}(C))}=\prod_{C^{\prime}\in{\mathscr{C}}_{j}}H(C^{\prime})\cong(S_{e_{1}}\wr\ldots\wr
S_{e_{j}})^{|{\mathscr{C}}_{j}(C)|}$, and the induced subgroup
$H(C)^{{\mathscr{C}}_{j}(C)}$ on ${\mathscr{C}}_{j}(C)$ is
$H(C)^{{\mathscr{C}}_{j}(C)}\cong H(C)/H(C)_{({\mathscr{C}}_{j}(C))}\cong
S_{e_{j+1}}\wr\ldots\wr S_{e_{i}}$, where
$|{\mathscr{C}}_{j}(C)|=\prod_{\ell=j+1}^{i}e_{\ell}$. Thus $H(C)\cong
H(C)_{({\mathscr{C}}_{j}(C))}\rtimes H(C)^{{\mathscr{C}}_{j}(C)}$, and
statement (b) follows.

(c) This follows by applying (b) to the case where $i=s$, since in this case
$C={\mathscr{P}}$ and $H(C)=G$. ∎

###  2.3. Array of a subset

The _array function_ of a nonempty subset $B$ of ${\mathscr{P}}$, with respect
to the chain $\bm{\mathscr{C}}$ of partitions, is the function
$\chi_{B}:\bigcup_{i=1}^{s}{\mathscr{C}}_{i}\rightarrow\mathbb{Z}_{\geq 0}$
defined by

(7) $\chi_{B}(C):=\left|B\cap C\right|\quad\text{for each
$C\in{\mathscr{C}}_{i}$ and each $i\in\\{1,\ldots,s\\}$.}$

In particular, for $i=s$ we have $\chi_{B}(C)=\chi_{B}({\mathscr{P}})=|B|$.
For brevity, and when there is no ambiguity, we shall frequently use the
notation $x_{C}:=\chi_{B}(C)$, or, more specifically,

(8) $x_{(\delta_{j})_{j>i}}:=\chi_{B}\big{(}C_{(\delta_{j})_{j>i}}\big{)}$

for $C_{(\delta_{j})_{j>i}}$ as in (3). For a subgroup $G\leq
S_{e_{1}}\wr\ldots\wr S_{e_{s}}$ which leaves invariant each partition in the
chain $\bm{\mathscr{C}}$, we say that two array functions $\chi_{B}$ and
$\chi_{B^{\prime}}$ are _equivalent under $G$_ if
$\chi_{B}^{g}=\chi_{B^{\prime}}$ for some $g\in G$, where

(9) $\chi_{B}^{g}(C):=\chi_{B}\big{(}C^{g^{-1}}\big{)}\quad\text{for all}\
C\in\bigcup_{i=1}^{s}{\mathscr{C}}_{i}.$

By [2, Lemma 2.4], if $G=S_{e_{1}}\wr\ldots\wr S_{e_{s}}$ then the $G$-orbit
of $B\subseteq{\mathscr{P}}$ consists of all
$B^{\prime}\subseteq{\mathscr{P}}$ whose array function $\chi_{B^{\prime}}$ is
equivalent to $\chi_{B}$ under $G$.

###  2.4. Block-transitive, chain-imprimitive $2$-designs

Consider a point-block incidence structure
${\mathscr{D}}=({\mathscr{P}},{\mathscr{B}})$, where
${\mathscr{P}}=\prod_{i=1}^{s}\mathbb{Z}_{e_{i}}$ and ${\mathscr{B}}=B^{G}$,
for some nonempty $B\subseteq{\mathscr{P}}$ and with $G=S_{e_{1}}\wr\ldots\wr
S_{e_{s}}$ leaving invariant an $s$-chain $\bm{\mathscr{C}}$ of partitions
${\mathscr{C}}_{i}$ of ${\mathscr{P}}$ as in (4). Then ${\mathscr{D}}$ is
$G$-block-transitive and $(G,s)$-chain-imprimitive. Theorem 1.3 in [2], stated
below, gives necessary and sufficient conditions on the array $\chi_{B}$ in
order for ${\mathscr{D}}$ to be a $2$-design. Recall that for
$C\in{\mathscr{C}}_{i-1}$, $C^{+}$ denotes the unique
${\mathscr{C}}_{i}$-class containing $C$.

######  Theorem 2.2.

[2, Theorem 1.3] Let $G=S_{e_{1}}\wr\ldots\wr S_{e_{s}}$ preserving $s$-chain
$\bm{\mathscr{C}}$ as in (4). Let
${\mathscr{D}}=({\mathscr{P}},{\mathscr{B}})$ where ${\mathscr{B}}=B^{G}$ for
some $k$-subset $B$ of ${\mathscr{P}}$. For any
$C\in\bigcup_{i=1}^{s}{\mathscr{C}}_{i}$ let $x_{C}=|B\cap C|$. Then
${\mathscr{D}}$ is a $2$-design if and only if

(10)
$\sum_{C\in{\mathscr{C}}_{1}}x_{C}\left(x_{C}-1\right)=\frac{k(k-1)}{v-1}(e_{1}-1)$

and

(11) $\text{for each
$i\in\\{2,\ldots,s\\}$,}\quad\sum_{C\in{\mathscr{C}}_{i-1}}x_{C}\left(x_{C^{+}}-x_{C}\right)=\frac{k(k-1)}{v-1}(e_{i}-1)\prod_{j\leq
i-1}e_{j}.$

Now, $C_{(\delta_{j})_{j>i-1}}^{+}=C_{(\delta_{j})_{j>i}}$ by (3). Thus, using
notation (8), we can rewrite condition (11) as

$\sum_{(\delta_{j})_{j>i-1}\in\prod_{j>i-1}\mathbb{Z}_{e_{j}}}x_{(\delta_{j})_{j>i-1}}\left(x_{(\delta_{j})_{j>i}}-x_{(\delta_{j})_{j>i-1}}\right)=\frac{k(k-1)}{v-1}(e_{i}-1)\prod_{j\leq
i-1}e_{j}.$

##  3\. Point subsets with uniform array functions

The theme of the paper is $G$-flag-transitive designs, where $G$ is chain-
imprimitive on the point set ${\mathscr{P}}$ with respect to a chain
$\bm{\mathscr{C}}$ as in (1). If $B$ is a block of such a design then the
setwise stabiliser $G_{B}$ is transitive on $B$. This means in particular
that, for each $i\in\\{0,\ldots,s\\}$, every ${\mathscr{C}}_{i}$-class $C$
which contains elements of $B$ must contain a constant number of elements of
$B$. In other words, there exist positive integers $y_{i}$, depending only on
$i$, such that for each $i\in\\{0,\ldots,s\\}$ and each
$C\in{\mathscr{C}}_{i}$, $\chi_{B}(C)=\big{|}B\cap
C\big{|}=x_{C}\in\\{0,y_{i}\\}$.

######  Definition 3.1.

A subset $B$ of ${\mathscr{P}}$ is _uniform_ relative to $\bm{\mathscr{C}}$,
with _uniform sequence_ $(y_{0},y_{1},\ldots,y_{s})$, if for each
$i\in\\{0,\ldots,s\\}$ and each $C\in{\mathscr{C}}_{i}(B)$, $\big{|}B\cap
C\big{|}=y_{i}$.

Note that for $i=0$ the class $C$ has a unique element, say $\delta$, and
$\chi_{B}(C)=0$ or $1$ according to whether $\delta\in B$ or $\delta\notin B$,
so $y_{0}=1$; while for $i=s$ we have $C={\mathscr{P}}$, and so
$\chi_{B}({\mathscr{P}})=|B|=k$ and thus $y_{s}=k$. Our next result derives
further restrictions on a uniform sequence and shows that all uniform subsets
with the same uniform sequence are in the same orbit under the stabiliser of
the chain $\bm{\mathscr{C}}$.

######  Lemma 3.2.

Let $s,e_{1},\dots,e_{s}$ be integers, all at least $2$, let
${\mathscr{P}}=\prod_{i=1}^{s}\mathbb{Z}_{e_{i}}$, let $\bm{\mathscr{C}}$ be a
chain of nontrivial partitions of ${\mathscr{P}}$, as in (1) with parts as in
(4), and chain stabiliser $G=S_{e_{1}}\wr\ldots\wr S_{e_{s}}$. For each
$i\in\\{0,\ldots,s\\}$, let $y_{i}\in\mathbb{Z}^{+}$.

  1. (a)

There exists a nonempty subset $B$ of ${\mathscr{P}}$ which is uniform
relative to $\bm{\mathscr{C}}$ with uniform sequence $(y_{0},\ldots,y_{s})$,
if and only if the following hold:

(i) $y_{0}=1$, $y_{s}=|B|$, and (ii) for all $i\in\\{1,\ldots,s\\}$,
$y_{i-1}\mid y_{i}$ with $\frac{y_{i}}{y_{i-1}}\leq e_{i}$.

  2. (b)

If conditions (i) and (ii) in part (a) hold, then the subset

(12) $B:=\left\\{(\delta_{i})_{i=1}^{s}\in{\mathscr{P}}\ \ \vline\ \
0\leq\delta_{i}\leq\frac{y_{i}}{y_{i-1}}-1\ \text{for }\ 1\leq i\leq
s\right\\}$

is uniform relative to $\bm{\mathscr{C}}$ with uniform sequence
$(y_{0},\ldots,y_{s})$. Moreover, a subset of ${\mathscr{P}}$ has these
properties if and only if it lies in the $G$-orbit of $B$.

###### Proof.

(a) Assume that there is a nonempty subset $B$ of ${\mathscr{P}}$ such that
$B$ is uniform relative to $\bm{\mathscr{C}}$ with uniform sequence
$(y_{0},\ldots,y_{s})$. Since all ${\mathscr{C}}_{0}$-classes have size $1$
and $B$ is non-empty, we have that $y_{0}=1$, and since
${\mathscr{C}}_{s}=\\{{\mathscr{P}}\\}$ we also have
$y_{s}=|B\cap{\mathscr{P}}|=|B|$. This proves condition (i). Let $1\leq i\leq
s$ and let $C$ be a ${\mathscr{C}}_{i}$-class intersecting $B$ non-trivially,
so that $x_{C}=y_{i}$. Consider the set ${\mathscr{C}}_{i-1}(C)$ of all
${\mathscr{C}}_{i-1}$-classes contained in $C$. For each
$C^{\prime}\in{\mathscr{C}}_{i-1}(C)$, $x_{C^{\prime}}\in\\{0,y_{i-1}\\}$, and
$x_{C}=\sum_{C^{\prime}\in{\mathscr{C}}_{i-1}(C)}x_{C^{\prime}}$. It follows
that $y_{i-1}\mid y_{i}$. Recall that $|{\mathscr{C}}_{i-1}(C)|=e_{i}$ so, if
$C^{\prime}\subset C$ is such that $x_{C^{\prime}}=y_{i-1}$, then we have
$x_{C}\leq e_{i}x_{C^{\prime}}$ and thus
$\frac{y_{i}}{y_{i-1}}=\frac{x_{C}}{x_{C^{\prime}}}\leq e_{i}$, which is
condition (ii). Therefore (i) and (ii) hold whenever some subset $B$ is
uniform relative to $\bm{\mathscr{C}}$ with uniform sequence
$(y_{0},\ldots,y_{s})$.

Conversely assume that conditions (i) and (ii) hold. Note that from condition
(ii), $0\leq\frac{y_{i}}{y_{i-1}}-1<e_{i}$. Hence if $(\delta_{i})_{i=1}^{s}$
satisfies the defining conditions for $B$ then
$(\delta_{i})_{i=1}^{s}\in\prod_{i=1}^{s}\mathbb{Z}_{e_{i}}={\mathscr{P}}$.
The array function $\chi_{B}$ of $B$ with respect to $\bm{\mathscr{C}}$ has
value
$\chi_{B}\big{(}C_{(\delta_{j})_{j>i}}\big{)}=\big{|}\big{\\{}(\varepsilon_{j})_{j=1}^{s}\in
B\ \big{|}\ \varepsilon_{j}=\delta_{j}\ \text{for all}\ j>i\big{\\}}\big{|}$,
which, using conditions (i) and (ii), can be computed as

(13)
$\chi_{B}\big{(}C_{(\delta_{j})_{j>i}}\big{)}=\begin{cases}\prod_{\ell=1}^{i}\frac{y_{\ell}}{y_{\ell-1}}=y_{i}&\text{if
$\delta_{j}<\frac{y_{j}}{y_{j-1}}$ for all $j>i$},\\\
0&\text{otherwise}.\end{cases}$

Therefore $B$ is uniform relative to $\bm{\mathscr{C}}$ with uniform sequence
$(y_{0},\ldots,y_{s})$. This completes the proof of statement (a), and also
proves the first part of statement (b).

(b) Let ${\mathscr{B}}=B^{G}$ with $B$ as in (12). Assume first that
$B^{\prime}\in{\mathscr{B}}$; so $B^{\prime}=B^{g}$ for some $g\in G$. By [2,
Lemma 2.5 (a)], the array function $\chi_{B^{\prime}}=\chi^{g}_{B}$ where
$\chi^{g}_{B}$ is as defined in (9). Hence, for each
$C\in\bigcup_{i=1}^{s}{\mathscr{C}}_{i}$, say $C\in{\mathscr{C}}_{i}$ so also
$C^{g^{-1}}\in{\mathscr{C}}_{i}$, we have
$\chi_{B^{\prime}}(C)=\chi_{B}^{g}(C)=\chi_{B}\big{(}C^{g^{-1}}\big{)}$ which
lies in $\\{0,y_{i}\\}$ since $B$ has uniform sequence $(y_{0},\ldots,y_{s})$.
Thus $B^{\prime}$ is also uniform relative to $\bm{\mathscr{C}}$ with uniform
sequence $(y_{0},\ldots,y_{s})$.

Conversely, let $B^{\prime}$ be a uniform subset of ${\mathscr{P}}$ relative
to $\bm{\mathscr{C}}$ with uniform sequence $(y_{0},\ldots,y_{s})$; we will
show that $B^{\prime}\in{\mathscr{B}}$.

_Claim. For every $j\in\\{0,\ldots,s-1\\}$, there exists $g\in G$ such that
${\mathscr{C}}_{j}(B)^{g}={\mathscr{C}}_{j}(B^{\prime})$. _

We prove the claim by induction on $\ell=s-j$, for $\ell\in\\{1,\ldots,s\\}$.
Suppose first that $\ell=1$, so that $j=s-1$. By (5),
$|{\mathscr{C}}_{s-1}(B)|$ is the number of classes of ${\mathscr{C}}_{s-1}$
that intersect $B$ nontrivially. Since $B$ and $B^{\prime}$ both have uniform
sequence $(y_{0},\ldots,y_{s})$, it follows from Definition 3.1 and Lemma 3.2
that $|B|=|B^{\prime}|=k=y_{s}$ and $|B\cap C|=y_{s-1}$ for
$C\in{\mathscr{C}}_{s-1}(B)$, while $|B^{\prime}\cap C|=y_{s-1}$ for
$C\in{\mathscr{C}}_{s-1}(B^{\prime})$. Thus
$|{\mathscr{C}}_{s-1}(B)|=|{\mathscr{C}}_{s-1}(B^{\prime})|=\frac{y_{s}}{y_{s-1}}$.
Then since $G$ induces $G^{{\mathscr{C}}_{s-1}}\cong S_{e_{s}}$ on
${\mathscr{C}}_{s-1}$ (Proposition 2.1(b)), there exists $g\in G$ such that
$g^{{\mathscr{C}}_{s-1}}$ maps ${\mathscr{C}}_{s-1}(B)$ to
${\mathscr{C}}_{s-1}(B^{\prime})$. Therefore the claim holds for $j=s-1$, that
is, for $\ell=1$.

Let $\ell\in\\{2,\ldots,s\\}$ and assume inductively that the claim holds for
$\ell-1$, that is, there exists $g\in G$ such that
${\mathscr{C}}_{j}(B)^{g}={\mathscr{C}}_{j}(B^{\prime})$ where $j:=s-\ell+1$.
Note that ${\mathscr{C}}_{j}(B)^{g}={\mathscr{C}}_{j}(B^{g})$ by (5). We
proved above that $B^{g}$ is uniform relative to $\bm{\mathscr{C}}$ with
uniform sequence $(y_{0},\ldots,y_{s})$. Thus for any
$C\in{\mathscr{C}}_{j}(B^{g})={\mathscr{C}}_{j}(B^{\prime})$ we have
$|{\mathscr{C}}_{j-1}(B^{g}\cap C)|=|{\mathscr{C}}_{j-1}(B^{\prime}\cap
C)|=\frac{y_{j}}{y_{j-1}}$. By Proposition 2.1(a), the group
$H(C)=G_{C}^{C}\cong S_{e_{1}}\wr\ldots\wr S_{e_{j}}$, and
$H(C)^{{\mathscr{C}}_{j-1}(C)}\cong S_{e_{j}}$. Hence there exists an element
$h_{C}\in H(C)$ such that ${\mathscr{C}}_{j-1}(B^{g}\cap
C)^{h_{C}}={\mathscr{C}}_{j-1}(B^{\prime}\cap C)$, and we may view $h_{C}$ as
a permutation of ${\mathscr{P}}$ (and hence an element of $G$) fixing
pointwise each of the parts of ${\mathscr{C}}_{j}\setminus\\{C\\}$. Such an
element $h_{C}$ exists for each of the parts $C\in{\mathscr{C}}_{j}(B^{g})$,
and the product of these elements $h_{C}$ over all
$C\in{\mathscr{C}}_{j}(B^{g})$ is an element of $G$ that maps
${\mathscr{C}}_{j-1}(B^{g})$ (which is the union over
$C\in{\mathscr{C}}_{j}(B^{g})$ of ${\mathscr{C}}_{j-1}(B^{g}\cap C)$) to
${\mathscr{C}}_{j-1}(B^{\prime})$. Since $j-1=s-\ell$, the claim is proved for
$\ell$.

Therefore the claim is proved by induction for all $\ell\in\\{1,\ldots,s\\}$.
In particular, for $\ell=s$, there exists $g\in G$ such that
${\mathscr{C}}_{0}(B^{g})={\mathscr{C}}_{0}(B^{\prime})$. Since
${\mathscr{C}}_{0}(B^{g})=\binom{B^{g}}{1}$ and
${\mathscr{C}}_{0}(B^{\prime})=\binom{B^{\prime}}{1}$, it follows that
$B^{g}=B^{\prime}$. Therefore $B^{\prime}\in{\mathscr{B}}$, which completes
the proof of part (b). ∎

Recall from Section 2.3 that the array functions $\chi_{B}$ and
$\chi_{B^{\prime}}$ of two subsets $B,B^{\prime}\subseteq{\mathscr{P}}$ are
equivalent under $G$ if $\chi_{B}^{g}=\chi_{B^{\prime}}$ for some $g\in G$.
Further, by [2, Lemma 2.5(b)], if $G=S_{e_{1}}\wr\ldots\wr S_{e_{s}}$ then the
$G$-orbit of $B\subseteq{\mathscr{P}}$ consists of all
$B^{\prime}\subseteq{\mathscr{P}}$ whose array function $\chi_{B^{\prime}}$ is
equivalent to $\chi_{B}$ under $G$.

######  Lemma 3.3.

Let $s,e_{1},\dots,e_{s}$ be integers, all at least $2$, let
${\mathscr{P}}=\prod_{i=1}^{s}\mathbb{Z}_{e_{i}}$, and let $\bm{\mathscr{C}}$
be a chain of nontrivial partitions of ${\mathscr{P}}$, as in (1) with chain
stabiliser $G=S_{e_{1}}\wr\ldots\wr S_{e_{s}}$. Let ${\mathscr{B}}=B^{G}$
where $B\subseteq{\mathscr{P}}$ is uniform relative to $\bm{\mathscr{C}}$ with
uniform sequence $(y_{0},\ldots,y_{s})$. Then

(14)
$|{\mathscr{B}}|=\prod_{j=1}^{s}\binom{e_{j}}{\frac{y_{j}}{y_{j-1}}}^{k/y_{j}}.$

###### Proof.

As in Proposition 2.1, we define the group $H(C)=G_{C}^{C}$ for any class
$C\in\bigcup_{j=1}^{s}{\mathscr{C}}_{j}$. Note that if
$C\in{\mathscr{C}}_{i}(B)$, then $B\cap C$ is uniform relative to
${\mathscr{C}}(C)$ with uniform sequence $(y_{0},\ldots,y_{i})$, since for
each $j\in\\{1,\ldots,i\\}$ and each $C^{\prime}\in{\mathscr{C}}_{j}(C)$,
$\chi_{B\cap C}(C^{\prime})=|B\cap C\cap C^{\prime}|=|B\cap
C^{\prime}|=\chi_{B}(C^{\prime})\in\\{0,y_{j}\\}$.

_Claim 1. For any $i\in\\{1,\ldots,s\\}$ and any $C\in{\mathscr{C}}_{i}(B)$,
the orbit $(B\cap C)^{H(C)}$ is the set of all subsets of $C$ that have
uniform sequence $(y_{0},\ldots,y_{i})$ relative to ${\mathscr{C}}(C)$._

If $i=1$ then by Proposition 2.1(a) the group $H(C)\cong S_{e_{1}}$. Hence
$(B\cap C)^{H(C)}$ is the set of all subsets of $C$ that have size $y_{1}$,
and thus have uniform sequence $(y_{0},y_{1})$ relative to ${\mathscr{C}}(C)$.
Therefore the claim holds for $i=1$. Now assume that $i\geq 2$. Using Lemma
3.2(b) (and by replacing ${\mathscr{P}}$ in the lemma with the set $C$, the
chain ${\mathscr{C}}$ with ${\mathscr{C}}(C)$, and the group $G$ with $H(C)$),
we deduce that the orbit $(B\cap C)^{H(C)}$ consists of all subsets of $C$
that have size $y_{i}$ and are uniform relative to ${\mathscr{C}}(C)$, with
uniform sequence $(y_{0},\ldots,y_{i})$. This proves Claim 1.

For $i\in\\{1,\ldots,s\\}$ let
$b_{i}:=\prod_{j=1}^{i}\binom{e_{j}}{y_{j}/y_{j-1}}^{y_{i}/y_{j}}$.

_Claim 2. For any $i\in\\{1,\ldots,s\\}$ and any
$C_{i}\in{\mathscr{C}}_{i}(B)$, $\big{|}(B\cap
C_{i})^{H(C_{i})}\big{|}=b_{i}$._

We prove this by induction on $i$. If $i=1$ then by Claim 1 the orbit $(B\cap
C_{1})^{H(C_{1})}$ is the set of all subsets of size $y_{1}$ of $C_{1}$; since
$|C_{1}|=e_{1}$, the number of such subsets is $\binom{e_{1}}{y_{1}}$.
Therefore, recalling that $y_{0}=1$, we have

$\big{|}(B\cap
C_{1})^{H(C_{1})}\big{|}=\binom{e_{1}}{y_{1}}=\prod_{j=1}^{1}\binom{e_{1}}{y_{1}/y_{0}}^{y_{1}/y_{1}}=b_{1},$

which shows that Claim 2 holds when $i=1$.

Suppose now that $i\geq 2$ and assume that $\big{|}(B\cap
C_{i-1})^{H(C_{i-1})}\big{|}=b_{i-1}$ for any
$C_{i-1}\in{\mathscr{C}}_{i-1}(B)$. By Claim 1, each subset
$B^{\prime}\in(B\cap C_{i})^{H(C_{i})}$ has size $y_{i}$, and is uniform
relative to ${\mathscr{C}}(C_{i})$ with uniform sequence
$(y_{0},\ldots,y_{i})$. The set $B^{\prime}$ is the disjoint union of subsets
$B^{\prime}\cap C_{i-1}$ for all $C_{i-1}\in{\mathscr{C}}_{i-1}(B^{\prime})$,
where, by the observation at the start of this proof, each subset
$B^{\prime}\cap C_{i-1}$ has size $y_{i-1}$ and is uniform relative to
${\mathscr{C}}(C_{i})$ with uniform sequence $(y_{0},\ldots,y_{i-1})$. Since
$B^{\prime}\cap C_{i-1}\subseteq C_{i-1}$, it follows from Claim 1 that
$B^{\prime}\cap C_{i-1}\in(B\cap C_{i-1})^{H(C_{i-1})}$. Conversely, note that
any subset $B^{\prime\prime}$ from an orbit $(B\cap C_{i-1})^{H(C_{i-1})}$,
for any $C_{i-1}\in{\mathscr{C}}_{i-1}(B\cap C_{i})$, is uniform relative to
${\mathscr{C}}(C_{i})$ with uniform sequence $(y_{0},\ldots,y_{i-1})$. The
number $|{\mathscr{C}}_{i-1}(B\cap C_{i})|$ of ${\mathscr{C}}_{i-1}$-classes
in $C_{i}$ that intersect $B$ non-trivially is $|{\mathscr{C}}_{i-1}(B\cap
C_{i})|=\frac{y_{i}}{y_{i-1}}$, so that
$\big{|}\bigcup_{C_{i-1}\in{\mathscr{C}}_{i-1}(B\cap
C_{i})}B^{\prime\prime}\big{|}=|B^{\prime\prime}|\cdot|{\mathscr{C}}_{i-1}(B\cap
C_{i})|=y_{i-1}\cdot y_{i}/y_{i-1}$. Thus, if we take one subset
$B^{\prime\prime}$ from each of the orbits $(B\cap C_{i-1})^{H(C_{i-1})}$,
where $C_{i-1}\in{\mathscr{C}}_{i-1}(B\cap C_{i})$, then the union of these
subsets $B^{\prime\prime}$ is a subset of $B\cap C_{i}$ of size $y_{i}$ that
is uniform relative to ${\mathscr{C}}(C_{i})$ with uniform sequence
$(y_{0},\ldots,y_{i})$. By Proposition 2.1(b) we deduce that

$H(C_{i})\cong\left(\prod_{C\in{\mathscr{C}}_{i-1}(C_{i})}H(C)\right)\rtimes
H(C_{i})^{{\mathscr{C}}_{i-1}(C_{i})}\cong\left(\prod_{C\in{\mathscr{C}}_{i-1}(C_{i})}H(C)\right)\rtimes
S_{e_{i}}.$

Since $B\cap C_{i}$ is the disjoint union over all
$C\in{\mathscr{C}}_{i-1}(B\cap C_{i})$ of subsets $B\cap C$, it follows that
the orbit $(B\cap C_{i})^{\prod_{C\in{\mathscr{C}}_{i-1}(C_{i})}H(C)}$ is the
disjoint union over all $C\in{\mathscr{C}}_{i-1}(B\cap C_{i})$ of orbits
$(B\cap C)^{H(C)}$. Hence

$(B\cap C_{i})^{H(C_{i})}=\left(\bigcup_{C\in{\mathscr{C}}_{i-1}(B\cap
C_{i})}(B\cap C)^{H(C)}\right)^{S_{e_{i}}}$

and therefore, since $|{\mathscr{C}}_{i-1}(B\cap C_{i})|=y_{i}/y_{i-1}$,

$\big{|}(B\cap C_{i})^{H(C_{i})}\big{|}=\big{|}(B\cap
C)^{H(C)}\big{|}^{y_{i}/y_{i-1}}\binom{e_{i}}{y_{i}/y_{i-1}}\quad\text{for any
$C\in{\mathscr{C}}_{i-1}(B\cap C_{i})$.}$

By the induction hypothesis $\big{|}(B\cap C)^{H(C)}\big{|}=b_{i-1}$.
Substituting this into the equation above, and applying the definition of
$b_{i-1}$, we obtain

$\big{|}(B\cap
C_{i})^{H(C_{i})}\big{|}=\Bigg{(}\prod_{j=1}^{i-1}\binom{e_{j}}{y_{j}/y_{j-1}}^{y_{i-1}/y_{j}}\Bigg{)}^{y_{i}/y_{i-1}}\cdot\binom{e_{i}}{y_{i}/y_{i-1}}=\prod_{j=1}^{i}\binom{e_{j}}{y_{j}/y_{j-1}}^{y_{i}/y_{j}}=b_{i}.$

Therefore Claim 2 holds by induction.

Finally, observe that $C_{s}={\mathscr{P}}$ and $G=H(C_{s})$, so that
${\mathscr{B}}=B^{G}=(B\cap C_{s})^{H(C_{s})}$. Therefore, by Claim 2 and
recalling that $y_{s}=k$, we obtain
$|{\mathscr{B}}|=b_{s}=\prod_{j=1}^{s}\binom{e_{j}}{y_{j}/y_{j-1}}^{k/y_{j}}$.
This completes the proof of the lemma. ∎

######  Lemma 3.4.

Let $s,e_{1},\dots,e_{s}$ be integers, all at least $2$, let
${\mathscr{P}}=\prod_{i=1}^{s}\mathbb{Z}_{e_{i}}$, and let $\bm{\mathscr{C}}$
be a chain of nontrivial partitions of ${\mathscr{P}}$, as in (1) with chain
stabiliser $G=S_{e_{1}}\wr\ldots\wr S_{e_{s}}$. Then for any subset
$B\subseteq{\mathscr{P}}$ which is uniform relative to $\bm{\mathscr{C}}$, the
setwise stabiliser $G_{B}$ of $B$ is transitive on $B$.

###### Proof.

Let $B\subseteq{\mathscr{P}}$ be uniform relative to $\bm{\mathscr{C}}$, with
uniform sequence $(y_{0},\ldots,y_{s})$. Since, by Lemma 3.2(b), every
$G$-orbit on uniform subsets of ${\mathscr{P}}$ with uniform sequence
$(y_{0},\ldots,y_{s})$ contains a subset $B$ of the form described in (12), it
is enough to prove the result for such a subset $B$. Observe that the subset
$B$ in (12) can be written as $B=E_{1}\times\ldots\times E_{s}$ where, for
$i\in\\{1,\ldots,s\\}$,

$E_{i}:=\left\\{\delta_{i}\in\mathbb{Z}_{e_{i}}\ \vline\
0\leq\delta_{i}<\frac{y_{i}}{y_{i-1}}\right\\}=\mathbb{Z}_{y_{i}/y_{i-1}}.$

For each $i\in\\{1,\ldots,s\\}$, the sets $B\cap C$, for all
$C\in{\mathscr{C}}_{i}(B)$, form a partition ${\mathscr{C}}_{i}^{B}$ of $B$,
and the partitions ${\mathscr{C}}_{i}^{B}$ form a chain
$\bm{\mathscr{C}}^{B}:\binom{B}{1}={\mathscr{C}}_{0}^{B}\prec{\mathscr{C}}_{1}^{B}\prec\ldots\prec{\mathscr{C}}_{s}^{B}=\\{B\\}$.
We note that some of the subsets $E_{i}$ may contain only a single element,
(if $y_{i-1}=y_{i}$), and hence some of these partitions may be equal. This
possible degeneracy does not affect our arguments. By Proposition 2.1(a) the
stabiliser in ${\mathrm{Sym}}(B)$ of this chain $\bm{\mathscr{C}}^{B}$ is the
group $L={\mathrm{Sym}}(E_{1})\wr\ldots\wr{\mathrm{Sym}}(E_{s})$. Identify
each group ${\mathrm{Sym}}(E_{i})$ with the subgroup of $S_{e_{i}}$ acting
naturally on $E_{i}$ and fixing $\mathbb{Z}_{e_{i}}\setminus E_{i}$ pointwise;
thus $L\leq G$ and in particular $L\leq G_{B}$. We shall show that $L$ is
transitive on $B$. By a result in permutation group theory (see, for instance,
[19, Lemma 5.4 (iii)]), the wreath product $X\wr Y$ of groups
$X\leq{\mathrm{Sym}}(\Gamma)$ and $Y\leq{\mathrm{Sym}}(\Delta)$ is transitive
in its natural action on $\Gamma\times\Delta$ if and only if $X$ is transitive
on $\Gamma$ and $Y$ is transitive on $\Delta$. (This is true even if one or
other of $\Gamma,\Delta$ has size $1$.) If $s=2$ then
$L={\mathrm{Sym}}(E_{1})\wr{\mathrm{Sym}}(E_{2})$, so by [19, Lemma 5.4 (iii)]
the group $L$ is transitive on $B$. Suppose that $s\geq 3$ and that
$K={\mathrm{Sym}}(E_{1})\wr\ldots\wr{\mathrm{Sym}}(E_{s-1})$ is transitive on
$E_{1}\times\ldots\times E_{s-1}$; then $L=K\wr{\mathrm{Sym}}(E_{s})$ and
again by [19, Lemma 5.4 (iii)] the group $L$ is transitive on
$B=(E_{1}\times\ldots\times E_{s-1})\times E_{s}$. Therefore, by induction,
$L$ is transitive on $B$. It follows that $G_{B}$ is transitive on $B$,
completing the proof. ∎

A $1$-design is a point-block incidence structure in which every point lies in
a constant number of blocks. Clearly, any point-block incidence structure
which admits a point-transitive automorphism group is a $1$-design.

######  Corollary 3.5.

Let $s,e_{1},\dots,e_{s}$ be integers, all at least $2$, let
${\mathscr{P}}=\prod_{i=1}^{s}\mathbb{Z}_{e_{i}}$, and let $\bm{\mathscr{C}}$
be a chain of nontrivial partitions of ${\mathscr{P}}$, as in (1) with chain
stabiliser $G=S_{e_{1}}\wr\ldots\wr S_{e_{s}}$. Then for any subset
$B\subseteq{\mathscr{P}}$ which is uniform relative to $\bm{\mathscr{C}}$ with
uniform sequence $(y_{0},\ldots,y_{s})$, the point-block incidence structure
${\mathscr{D}}=({\mathscr{P}},B^{G})$ is a $G$-flag-transitive $1$-design.

###### Proof.

The conclusion that ${\mathscr{D}}$ is a $1$-design follows from the
transitivity of $G$ on ${\mathscr{P}}$. Flag-transitivity follows from Lemma
3.4. ∎

##  4\. A general construction for flag-transitive, $s$-chain-imprimitive
$2$-designs

In this section we give a general construction for $2$-designs based on the
existence of a sequence of integers $y_{i}$ as in (2) with the properties
(FT1) and (FT2) of Theorem 1.2. We first deduce from these properties a
divisibility condition for these numbers $y_{i}$.

######  Lemma 4.1.

Let $s,k,e_{1},\dots,e_{s}$ be integers, all at least $2$ such that $k<v$
where $v=\prod_{i=1}^{s}e_{i}$. Let $d=\gcd(e_{1}-1,\dots,e_{s}-1)$, let
$y_{0}=1$, and for $1\leq i\leq s$ let

$y_{i}=1+\frac{k-1}{v-1}\left(\left(\prod_{j=1}^{i}e_{j}\right)-1\right).$

Assume that $v-1$ divides $(k-1)\cdot d$.

  1. (a)

The number $u:=\frac{(k-1)d}{v-1}\in\mathbb{Z}$ and each $y_{i}$ is a positive
integer coprime to $u$. Moreover $y_{1}-y_{0}=\frac{k-1}{v-1}(e_{1}-1)$, and
for $2\leq i\leq s$,
$y_{i}-y_{i-1}=\frac{k-1}{v-1}(e_{i}-1)\prod_{j=1}^{i-1}e_{j}$.

  2. (b)

If, in addition, for each $i\in\\{1,\ldots,s\\}$ the integer $y_{i-1}$ divides
$(e_{i}-1)\left(\prod_{j=1}^{i-1}e_{j}\right)/d$, then for each
$i\in\\{1,\dots,s\\}$, $y_{i-1}$ divides $y_{i}$ and
$1<\frac{y_{i}}{y_{i-1}}<e_{i}$.

###### Proof.

For $0\leq i\leq s$, let $c_{i}=\prod_{j=0}^{i}e_{j}$, with $e_{0}=1$. Then
for each $i=1,\dots,s$,

$c_{i}-1=c_{i-1}(e_{i}-1)+\dots+c_{1}(e_{2}-1)+(e_{1}-1),$

and it follows that $d=\gcd(e_{1}-1,\dots,e_{s}-1)$ divides $c_{i}-1$; this
holds also for $i=0$ since $c_{0}=1$. By assumption $v-1$ divides $(k-1)\cdot
d$, so $u:=\frac{(k-1)d}{v-1}\in\mathbb{Z}$. Thus we can write $y_{i}$ as

(15) $y_{i}=1+\frac{u}{d}\cdot(c_{i}-1)\quad\text{for $0\leq i\leq s$},$

proving that $y_{i}\in\mathbb{Z}^{+}$ and $y_{i}$ is coprime to $u$. Then, for
$1\leq i\leq s$, using this equation, and noting that $c_{i}=c_{i-1}e_{i}$, we
have

$y_{i}-y_{i-1}=\frac{u}{d}\cdot(c_{i}-c_{i-1})=\frac{u}{d}\cdot(e_{i}-1)c_{i-1}=\frac{k-1}{v-1}(e_{i}-1)\prod_{j=1}^{i-1}e_{j}.$

This proves part (a), noting that the product is ‘empty’ if $i=1$.

Now assume, in addition, that for each $i\in\\{1,\ldots,s\\}$ the integer
$y_{i-1}$ divides
$(e_{i}-1)\left(\prod_{j=1}^{i-1}e_{j}\right)/d=\frac{(e_{i}-1)c_{i-1}}{d}$.
Thus, since $u\in\mathbb{Z}$, it follows that $y_{i-1}$ divides
$y_{i}-y_{i-1}$, and therefore $y_{i-1}$ divides $y_{i}$. Since $e_{i}\geq 2$
for all $i$, the $c_{i}$’s are strictly increasing. Then by (15), the
$y_{i}$’s are also strictly increasing, and so $\frac{y_{i}}{y_{i-1}}>1$.
Since $y_{i-1}=1+u(c_{i-1}-1)/d$, by (15), we get that

$y_{i-1}e_{i}=e_{i}+u\cdot\frac{(c_{i-1}-1)e_{i}}{d}=e_{i}\left(1-\frac{u}{d}\right)+u\cdot\frac{c_{i-1}e_{i}}{d}=e_{i}\left(1-\frac{u}{d}\right)+u\cdot\frac{c_{i}}{d}.$

Now $k<v$ implies that $\frac{u}{d}=\frac{k-1}{v-1}<1$, so $1-\frac{u}{d}>0$.
Then, since $e_{i}\geq 2$,

$e_{i}\left(1-\frac{u}{d}\right)+u\cdot\frac{c_{i}}{d}>\left(1-\frac{u}{d}\right)+u\cdot\frac{c_{i}}{d}=1+u\cdot\frac{c_{i}-1}{d}=y_{i}.$

Therefore $\frac{y_{i}}{y_{i-1}}<e_{i}$. This proves part (b). ∎

Now we give the general design construction.

######  Construction 4.2.

Let $s\geq 2$ and $e_{1},\ldots,e_{s}\geq 2$ be integers,
$v:=\prod_{i=1}^{s}e_{i}$, $d:=\gcd(e_{1}-1,\ldots,e_{s}-1)$, and let $k$ be
an integer with $1<k<v$. Let $y_{0}:=1$ and for $i\in\\{1,\ldots,s\\}$ let
$y_{i}$ be as in (2), that is,

$y_{i}:=1+\frac{k-1}{v-1}\left(\left(\prod_{1\leq j\leq
i}e_{j}\right)-1\right)\quad\text{for $i\in\\{1,\ldots,s\\}$}.$

Assume that properties (FT1) and (FT2) in Theorem 1.2 both hold, that is,
$v-1$ divides $(k-1)\cdot d$, and for each $i\in\\{1,\ldots,s-1\\}$, the
integer $y_{i}$ divides $(e_{i+1}-1)\left(\prod_{j=1}^{i}e_{j}\right)/d$. Let
${\mathscr{P}}=\prod_{i=1}^{s}\mathbb{Z}_{e_{i}}$, and let
$G=S_{e_{1}}\wr\ldots\wr S_{e_{s}}$, the stabiliser of the partition chain
$\bm{\mathscr{C}}$ as in (1) and (4). Also let
$\mathbf{e}:=(e_{1},\ldots,e_{s})$ and let $\mathscr{D}^{\rm
ft}(\mathbf{e};k)=({\mathscr{P}},B^{G})$, where $B$ is as in (12), that is,

$B=\bigg{\\{}(\delta_{1},\ldots,\delta_{s})\ \bigg{|}\
0\leq\delta_{i}\leq\frac{y_{i}}{y_{i-1}}-1\ \text{for $1\leq i\leq
s$}\bigg{\\}}.$

######  Proposition 4.3.

With the hypotheses of Construction 4.2,

  1. (a)

the set $B$ has cardinality $k$, and

  2. (b)

$\mathscr{D}^{\rm ft}(\mathbf{e};k)$ is a $G$-flag-transitive, $(G,s)$-chain-
imprimitive $2$-$(v,k,\lambda)$ design, with $\lambda=\frac{bk(k-1)}{v(v-1)}$
where $b$ is as given in (14) in Lemma _3.3_.

###### Proof.

By Lemma 4.1, for each $i\in\\{1,\ldots,s\\}$, $y_{i}$ is an integer,
$y_{i-1}$ divides $y_{i}$, and $1<\frac{y_{i}}{y_{i-1}}\leq e_{i}$. Therefore
the set $B$ is uniform relative to ${\mathscr{C}}$ with uniform sequence
$(y_{0},\ldots,y_{s})$ by Lemma 3.2. In particular $|B|=y_{s}=k$.

By Corollary 3.5, ${\mathscr{D}}=\mathscr{D}^{\rm ft}(\mathbf{e};k)$ is a
$G$-flag-transitive $1$-design, and by construction ${\mathscr{D}}$ is
$(G,s)$-chain-imprimitive. It remains to prove that ${\mathscr{D}}$ is a
$2$-design, and we do this by verifying conditions (10) and (11) of Theorem
2.2 for $i\in\\{2,\ldots,s\\}$. Since $B$ has uniform sequence
$(y_{0},\ldots,y_{s})$, we have that $x_{C}=y_{i}$ for
$C\in{\mathscr{C}}_{i}(B)$, and $x_{C}=0$ for
$C\in{\mathscr{C}}_{i}\setminus{\mathscr{C}}_{i}(B)$. The set $B$ is the
disjoint union of intersections $B\cap C$ for all $C\in{\mathscr{C}}_{i}(B)$,
so that
$\sum_{C\in{\mathscr{C}}_{i}(B)}y_{i}=\sum_{C\in{\mathscr{C}}_{i}(B)}|B\cap
C|=|B|=k$. In particular, for $i=1$,

$\sum_{C\in{\mathscr{C}}_{1}}x_{C}(x_{C}-1)=\sum_{C\in{\mathscr{C}}_{1}(B)}y_{1}(y_{1}-1)=\left(\sum_{C\in{\mathscr{C}}_{1}(B)}y_{1}\right)(y_{1}-1)=k(y_{1}-1).$

Condition (10) follows since, by the definition of $y_{1}$, we have
$y_{1}-1=\frac{k-1}{v-1}(e_{1}-1)$. Let $i\in\\{2,\ldots,s\\}$ and let $C^{+}$
be the unique ${\mathscr{C}}_{i}$-class containing $C\in{\mathscr{C}}_{i-1}$.
As noted above, for $C\in{\mathscr{C}}_{i-1}(B)$ we have $x_{C}=y_{i-1}$ and
$x_{C^{+}}=y_{i}$, and for $C\not\in{\mathscr{C}}_{i-1}(B)$ we have $x_{C}=0$.
Thus

$\sum_{C\in{\mathscr{C}}_{i-1}}x_{C}\left(x_{C^{+}}-x_{C}\right)=\sum_{C\in{\mathscr{C}}_{i-1}(B)}y_{i-1}\left(y_{i}-y_{i-1}\right)=\left(\sum_{C\in{\mathscr{C}}_{i-1}(B)}y_{i-1}\right)\cdot\left(y_{i}-y_{i-1}\right)=k(y_{i}-y_{i-1}).$

By Lemma 4.1(a),
$y_{i}-y_{i-1}=\frac{k-1}{v-1}(e_{i}-1)\prod_{j=1}^{i-1}e_{j}$, and condition
(11) follows. Therefore, by Theorem 2.2,
${\mathscr{D}}=({\mathscr{P}},{\mathscr{B}})$ is a $2$-design. The formula for
$\lambda$ follows immediately since we determined $b=|B^{G}|$ in Lemma _3.3_ ,
completing the proof. ∎

##  5\. Proof of Theorem 1.2

Let $s,k,e_{1},\ldots,e_{s}$ be integers, all at least $2$, such that $k<v$,
where $v:=\prod_{i=1}^{s}e_{i}$. Let
${\mathscr{P}}=\prod_{i=1}^{s}\mathbb{Z}_{e_{i}}$, of size $v$, and let
$W=S_{e_{1}}\wr\ldots\wr S_{e_{s}}$, the stabiliser of the partition chain
$\bm{\mathscr{C}}$ as in (1) and (4). So for $1\leq i\leq s$ each class of
${\mathscr{C}}_{i}$ contains $e_{i}$ classes of ${\mathscr{C}}_{i-1}$. Let
$d:=\gcd(e_{1}-1,\ldots,e_{s}-1)$.

Suppose first that $y_{1},\dots,y_{s}$ are as in (2) and that conditions (10)
and (11) of Theorem 2.2 hold. Then by Proposition 4.3, the design
$\mathscr{D}^{\rm ft}(\mathbf{e};k)$ of Construction 4.2 is a $(G,s)$-chain-
imprimitive, $G$-flag-transitive $2$-design, with blocks of size $k$, for the
group $G=W$.

Now suppose conversely that ${\mathscr{D}}=({\mathscr{P}},{\mathscr{B}})$ is a
$(G,s)$-chain-imprimitive, $G$-flag-transitive $2$-design, with blocks of size
$k$, for some group $G\leq W$ (so $G$ leaves invariant the chain
$\bm{\mathscr{C}}$). As discussed after the statement of Theorem 1.2,
$({\mathscr{P}},B^{W})$ is also a $2$-design, for $B\in{\mathscr{B}}$, so in
order to verify that conditions (FT1) and (FT2) hold for the constants $y_{i}$
as in (2), we may assume that $G=W$. Since ${\mathscr{D}}$ is $G$-flag-
transitive, the setwise stabiliser $G_{B}$ of the block $B$ is transitive on
$B$. Thus for each $i\in\\{0,\ldots,s\\}$, since $G_{B}$ leaves
${\mathscr{C}}_{i}$ invariant, every ${\mathscr{C}}_{i}$-class which contains
elements of $B$ must contain a constant number of elements of $B$. That is,
$B$ is uniform relative to $\bm{\mathscr{C}}$ for some uniform sequence
$(y^{\prime}_{0},\ldots,y^{\prime}_{s})$.

_Claim 1: $(y^{\prime}_{0},\ldots,y^{\prime}_{s})=(y_{0},\ldots,y_{s})$. _

By (2) and Lemma 3.2(a), we have $y^{\prime}_{0}=1=y_{0}$ and
$y^{\prime}_{s}=k=y_{s}$. So assume that $1\leq i<s$. Since $B$ is the
disjoint union of all the intersections $B\cap C$ with
$C\in{\mathscr{C}}_{i}(B)$, we have
$\sum_{C\in{\mathscr{C}}_{i}(B)}y^{\prime}_{i}=\sum_{C\in{\mathscr{C}}_{i}(B)}|B\cap
C|=|B|=k$. Also, for any $C\in{\mathscr{C}}_{i}$, we have
$x_{C}=y^{\prime}_{i}$ if $C\in{\mathscr{C}}_{i}(B)$, and $x_{C}=0$ otherwise;
and since ${\mathscr{D}}$ is a $2$-design, it follows from Theorem 2.2 that
conditions (10) and (11) in that theorem hold. We will show by induction that
$y^{\prime}_{i}=y_{i}$ for each $i\in\\{1,\dots,s\\}$.

By (10) we have
$\sum_{C\in{\mathscr{C}}_{1}}x_{C}(x_{C}-1)=\frac{k(k-1)}{v-1}(e_{1}-1)$,
while

$\sum_{C\in{\mathscr{C}}_{1}}x_{C}(x_{C}-1)=\sum_{C\in{\mathscr{C}}_{1}(B)}y^{\prime}_{1}(y^{\prime}_{1}-1)=k(y^{\prime}_{1}-1).$

Hence, $y^{\prime}_{1}=1+\frac{k-1}{v-1}(e_{1}-1)$ which equals $y_{1}$ by
(2). Suppose now that $2\leq i\leq s$ and $y^{\prime}_{i-1}=y_{i-1}$. Note
that $C\in{\mathscr{C}}_{i-1}(B)$ implies that $C^{+}\in{\mathscr{C}}_{i}(B)$,
where $C^{+}$ is the unique ${\mathscr{C}}_{i}$ class containing $C$. Thus,
for any $C\in{\mathscr{C}}_{i-1}$, $x_{C}=y^{\prime}_{i-1}$ and
$x_{C^{+}}=y^{\prime}_{i}$ whenever $C\in{\mathscr{C}}_{i-1}(B)$, and
$x_{C}=0$ otherwise, so

$\sum_{C\in{\mathscr{C}}_{i-1}}x_{C}(x_{C^{+}}-x_{C})=\sum_{C\in{\mathscr{C}}_{i-1}(B)}y^{\prime}_{i-1}(y^{\prime}_{i}-y^{\prime}_{i-1})=\left(\sum_{C\in{\mathscr{C}}_{i-1}(B)}y^{\prime}_{i-1}\right)(y^{\prime}_{i}-y^{\prime}_{i-1})=k(y^{\prime}_{i}-y^{\prime}_{i-1}).$

On the other hand, by (11),
$\sum_{C\in{\mathscr{C}}_{i-1}}x_{C}(x_{C^{+}}-x_{C})=\frac{k(k-1)}{v-1}(e_{i}-1)\prod_{j=1}^{i-1}e_{j}$.
Hence

(16) $y^{\prime}_{i}=y^{\prime}_{i-1}+\frac{k-1}{v-1}(e_{i}-1)\prod_{j\leq
i-1}e_{j}.$

By the induction hypothesis, we have

$\displaystyle y^{\prime}_{i}$
$\displaystyle=y_{i-1}+\frac{k-1}{v-1}(e_{i}-1)\prod_{j\leq i-1}e_{j}$
$\displaystyle=1+\frac{k-1}{v-1}\left(\Bigg{(}\prod_{j\leq
i-1}e_{j}\Bigg{)}-1+(e_{i}-1)\prod_{j\leq i-1}e_{j}\right)$
$\displaystyle=1+\frac{k-1}{v-1}\left(\Bigg{(}\prod_{j\leq
i}e_{j}\Bigg{)}-1\right)=y_{i}.$

Thus Claim 1 is proved.

_Claim 2: Condition(FT1) of Theorem 1.2 holds._ Observe that in equation (16)
(which holds for all $i\in\\{1,\dots,s\\}$) the numbers
$y^{\prime}_{i},y^{\prime}_{i-1}\in\mathbb{Z}$, so also
$\frac{k-1}{v-1}(e_{i}-1)\prod_{j\leq i-1}e_{j}\in\mathbb{Z}$. Now
$v=\prod_{j=1}^{s}e_{j}$ so $\gcd\big{(}v-1,\ \prod_{j\leq
i-1}e_{j}\big{)}=1$. Hence $v-1$ divides $(k-1)(e_{i}-1)$, and this is true
for all $i\in\\{1,\ldots,s\\}$. Therefore $v-1$ divides $(k-1)\cdot d$ where
$d=\gcd(e_{1}-1,\ldots,e_{s}-1)$, proving (FT1).

_Claim 3: Condition(FT2) of Theorem 1.2 holds._ Let $i\in\\{1,\dots,s-1\\}$,
and let $u:=\frac{(k-1)d}{v-1}$, so $u\in\mathbb{Z}^{+}$ by Claim 2. By Claims
1 and 2, the hypotheses of Lemma 4.1(a) hold, and this result implies that all
the $y_{j}$ are positive integers coprime to $u$, and that

$y_{i+1}-y_{i}=u\cdot\frac{e_{i+1}-1}{d}\cdot\prod_{j\leq i}e_{j}.$

Since $B$ has uniform sequence $(y_{0},\ldots,y_{s})$ relative to
$\bm{\mathscr{C}}$ by Claim 1, the parameter $y_{i}$ divides $y_{i+1}$ by
Lemma 3.2(a), and hence $y_{i}$ divides $y_{i+1}-y_{i}$. So $y_{i}$ divides
$u\cdot\frac{e_{i+1}-1}{d}\cdot\prod_{j\leq i}e_{j}$, and since $y_{i}$ is
coprime to $u$, $y_{i}$ divides $\frac{e_{i+1}-1}{d}\cdot\prod_{j\leq
i}e_{j}$. This proves condition (FT2), so Claim 3 is proved.

This proves the main assertion of Theorem 1.2, and the final assertion follows
from Claim 1. Thus Theorem 1.2 is proved.

######  Remark 5.1.

The discussion above allows us to conclude that, for given
$s,e_{1},\dots,e_{s},v,k$, such that conditions (FT1) and (FT2) of Theorem 1.2
hold for the constants $y_{i}$ in (2), there is a _unique_ $G$-flag-transitive
$(G,s)$-chain-imprimitive $2$-design with block size $k$, for $G$ the full
stabiliser of the $s$-chain $\bm{\mathscr{C}}$ in (1), namely the design
$\mathscr{D}^{\rm ft}(\mathbf{e};k)$ in Construction 4.2. To see this note
that, if ${\mathscr{D}}=({\mathscr{P}},B^{G})$ is such a design, then Claim 1
above shows that $B$ is uniform relative to $\bm{\mathscr{C}}$ with uniform
sequence $(y_{0},\ldots,y_{s})$, where each $y_{i}$ is as defined in (2). It
follows from Lemma 3.2(b) that ${\mathscr{B}}$ consists of all subsets of
${\mathscr{P}}$ which are uniform relative to $\bm{\mathscr{C}}$ with uniform
sequence $(y_{0},\ldots,y_{s})$. In particular,
${\mathscr{D}}=\mathscr{D}^{\rm ft}(\mathbf{e};y_{s})=\mathscr{D}^{\rm
ft}(\mathbf{e};k)$ is the unique such $2$-design.

##  6\. Explicit constructions for flag-transitive, $s$-chain-imprimitive
$2$-designs

Note that, although Construction 4.2 gives a general construction for flag-
transitive, $s$-chain-imprimitive $2$-designs based on a collection of
integers $y_{i}$ satisfying certain conditions, we need to produce integers
$y_{i}$ with the required conditions before we can guarantee existence of such
$2$-designs. We do this in Construction 6.1, where we give explicit parameters
$e_{1},\ldots,e_{s}$, and $k$ such that the integers $y_{i}$ satisfy the
hypotheses of Construction 4.2.

######  Construction 6.1.

Let $s,d$ be integers, both at least $2$, and define integers
$e_{1},\ldots,e_{s},k$ by

(17) $e_{1}=d+1,\quad e_{i}=d+\prod_{j\leq i-1}e_{j},\quad\text{for $2\leq
i\leq s$, and}\quad k:=1+\frac{v-1}{d}\text{ where }v=\prod_{j\leq s}e_{j}.$

Let ${\mathscr{P}}:=\prod_{i=1}^{s}\mathbb{Z}_{e_{i}}$, and let
$G=S_{e_{1}}\wr\ldots\wr S_{e_{s}}$, the stabiliser of the partition chain
$\bm{\mathscr{C}}$ as in (1) with partitions as in (4). For
$i\in\\{1,\dots,s\\}$, let $y_{i}$ be as in (2), let $B$ be as in (12), and
let $\mathscr{D}^{\rm ft}(\mathbf{e};k)$ be as in Construction 4.2, where
$\mathbf{e}=(e_{1},\ldots,e_{s})$.

######  Proposition 6.2.

The incidence structure $\mathscr{D}^{\rm ft}(\mathbf{e};k)$ of Construction
_6.1_ is a $G$-flag-transitive, $(G,s)$-chain-imprimitive $2$-$(v,k,\lambda)$
design with $G=S_{e_{1}}\wr\ldots\wr S_{e_{s}}$ and $\lambda=\frac{bk}{vd}$,
where $b=\prod_{i=1}^{s}\binom{e_{i}}{y_{i}/y_{i-1}}^{k/y_{i}}$ with
$y_{i}=\frac{e_{i+1}-1}{d}$ for all $i\in\\{1,\ldots,s-1\\}$, and $y_{s}=k$.

###### Proof.

First we prove that $d$ is equal to $d^{\prime}:=\gcd(e_{1}-1,\dots,e_{s}-1)$.
We show inductively that $d$ divides $e_{i}-1$ for each $i$: by (17) we have
$d=e_{1}-1$ so this holds for $i=1$, and if $1<i\leq s$ and $d$ divides
$e_{j}-1$ for all $j<i$, then by (17), $e_{i}\equiv d+1\equiv 1\pmod{d}$, so
$d$ also divides $e_{i}-1$. Thus $d$ divides $d^{\prime}$. On the other hand
$d^{\prime}\leq e_{1}-1=d$ and hence $d=d^{\prime}$.

Thus to prove that $\mathscr{D}^{\rm ft}(\mathbf{e};k)$ is a $2$-design, it is
sufficient to verify that properties (FT1) and (FT2) in Theorem 1.2 hold,
since then all the assertions will follow from Proposition 4.3. First we note
that, by (17), $v-1=(k-1)\cdot d$, and hence $v-1$ divides $(k-1)\cdot d$,
proving property (FT1). The second property (FT2) involves the $y_{i}$ for
$1\leq i<s$. Since $\frac{k-1}{v-1}=\frac{1}{d}$ we have, for each
$i\in\\{1,\ldots,s-1\\}$, that $\prod_{j\leq i}e_{j}=e_{i+1}-d$, so by (2),

$\displaystyle y_{i}=1+\frac{k-1}{v-1}\left(\left(\prod_{j\leq
i}e_{j}\right)-1\right)=1+\frac{1}{d}\left(\left(e_{i+1}-d\right)-1\right)=\frac{d+(e_{i+1}-d-1)}{d}=\frac{e_{i+1}-1}{d}.$

This implies firstly that $y_{i}$ is a positive integer, since $d$ divides
$e_{i+1}-1$. Also it implies that $y_{i}$ divides
$(e_{i+1}-1)\left(\prod_{j\leq i}e_{j}\right)/d$. Thus property (FT2) holds,
and therefore $\mathscr{D}^{\rm ft}(\mathbf{e},k)$ is a $2$-design.

Recall that $\lambda=\frac{bk(k-1)}{v(v-1)}=\frac{bk}{vd}$, where
$b:=|{\mathscr{B}}|$ is computed using the formula (14) in Lemma 3.3. Recall
$y_{i}=\frac{e_{i+1}-1}{d}$ for $i<s$ and $y_{s}=k$. This completes the proof.
∎

Having proved Proposition 6.2, it is now simple to deduce Theorem 1.1.

###### Proof of Theorem 1.1.

For each $s\geq 2$ we have infinitely many choices for the integer $d\geq 2$,
and hence, by Proposition 6.2, Construction 6.1 yields infinitely many flag-
transitive, $s$-chain-imprimitive $2$-designs, proving Theorem 1.1. ∎

###  6.1. Further examples

For the case where $s=3$, by Theorem 1.2, a flag-transitive, $3$-chain-
imprimitive $2$-design $\mathscr{D}^{\rm ft}(\mathbf{e};k)$ as in Construction
4.2 exists if and only if the parameters $\mathbf{e}=(e_{1},e_{2},e_{3}),$ and
$k$ satisfy the following conditions:

  1. (1)

$v-1=e_{1}e_{2}e_{3}-1$ divides $(k-1)\cdot d$, where
$d=\gcd(e_{1}-1,e_{2}-1,e_{3}-1)$;

  2. (2)

$1+\frac{k-1}{e_{1}e_{2}e_{3}-1}(e_{1}-1)$ divides $\frac{(e_{2}-1)e_{1}}{d}$;

  3. (3)

$1+\frac{k-1}{e_{1}e_{2}e_{3}-1}(e_{1}e_{2}-1)$ divides
$\frac{(e_{3}-1)e_{1}e_{2}}{d}$.

For each solution $\mathbf{e}=(e_{1},e_{2},e_{3}),k$, we obtain an example
$\mathscr{D}^{\rm ft}(\mathbf{e};k)$ from the general construction in
Construction 4.2. An exhaustive search using Magma [4] produced all possible
parameters satisfying these three conditions for which $e_{1},e_{2},e_{3}\leq
50$. These are listed in Table 1 and we indicate in boldface font the
parameter sets which are obtained using Construction 6.1. Thus we have:

######  Theorem 6.3.

For $e_{1},e_{2},e_{3}\leq 50$ and $\mathbf{e}:=(e_{1},e_{2},e_{3}),$ there
exists $k\in\mathbb{Z}$ such that the conditions of Construction  _4.2_ are
satisfied and $\mathscr{D}^{\rm ft}(\mathbf{e};k)$ is a flag-transitive
$3$-chain-imprimitive $2$-design if and only if $(e_{1},e_{2},e_{3},k)$ are as
listed in Table 1.

$e_{1}$ | $e_{2}$ | $e_{3}$ | $v=e_{1}e_{2}e_{3}$ | $k$ | $y_{1}$ | $y_{2}$
---|---|---|---|---|---|---
$\mathbf{3}$ | $\mathbf{5}$ | $\mathbf{17}$ | $\mathbf{255}$ | $\mathbf{128}$ | $\mathbf{2}$ | $\mathbf{8}$
$3$ | $5$ | $33$ | $495$ | $248$ | $2$ | $8$
$3$ | $5$ | $49$ | $735$ | $368$ | $2$ | $8$
$3$ | $9$ | $29$ | $783$ | $392$ | $2$ | $14$
$3$ | $13$ | $41$ | $1599$ | $800$ | $2$ | $20$
$4$ | $4$ | $10$ | $160$ | $54$ | $2$ | $6$
$4$ | $4$ | $19$ | $304$ | $102$ | $2$ | $6$
$4$ | $4$ | $28$ | $448$ | $150$ | $2$ | $6$
$4$ | $4$ | $37$ | $592$ | $198$ | $2$ | $6$
$4$ | $4$ | $46$ | $736$ | $246$ | $2$ | $6$
$4$ | $7$ | $16$ | $448$ | $150$ | $2$ | $10$
$\mathbf{4}$ | $\mathbf{7}$ | $\mathbf{31}$ | $\mathbf{868}$ | $\mathbf{290}$ | $\mathbf{2}$ | $\mathbf{10}$
$4$ | $7$ | $46$ | $1288$ | $430$ | $2$ | $10$
$4$ | $10$ | $22$ | $880$ | $294$ | $2$ | $14$
$4$ | $10$ | $43$ | $1720$ | $574$ | $2$ | $14$
$4$ | $13$ | $28$ | $1456$ | $486$ | $2$ | $18$
$4$ | $16$ | $34$ | $2176$ | $726$ | $2$ | $22$
$4$ | $19$ | $40$ | $3040$ | $1014$ | $2$ | $26$
$4$ | $22$ | $46$ | $4048$ | $1350$ | $2$ | $30$
$5$ | $7$ | $37$ | $1295$ | $648$ | $3$ | $18$
$5$ | $9$ | $17$ | $765$ | $192$ | $2$ | $12$
$5$ | $9$ | $33$ | $1485$ | $372$ | $2$ | $12$
$\mathbf{5}$ | $\mathbf{9}$ | $\mathbf{49}$ | $\mathbf{2205}$ | $\mathbf{552}$ | $\mathbf{2}$ | $\mathbf{12}$
$6$ | $6$ | $11$ | $396$ | $80$ | $2$ | $8$
$6$ | $6$ | $21$ | $756$ | $152$ | $2$ | $8$
$6$ | $6$ | $26$ | $936$ | $375$ | $3$ | $15$
$6$ | $6$ | $31$ | $1116$ | $224$ | $2$ | $8$
$6$ | $6$ | $41$ | $1476$ | $296$ | $2$ | $8$
$6$ | $11$ | $36$ | $2376$ | $476$ | $2$ | $14$

$e_{1}$ | $e_{2}$ | $e_{3}$ | $v=e_{1}e_{2}e_{3}$ | $k$ | $y_{1}$ | $y_{2}$
---|---|---|---|---|---|---
$6$ | $11$ | $46$ | $3036$ | $1215$ | $3$ | $27$
$6$ | $16$ | $26$ | $2496$ | $500$ | $2$ | $20$
$6$ | $26$ | $41$ | $6396$ | $1280$ | $2$ | $32$
$7$ | $10$ | $37$ | $2590$ | $864$ | $3$ | $24$
$7$ | $25$ | $37$ | $6475$ | $1080$ | $2$ | $30$
$8$ | $8$ | $36$ | $2304$ | $330$ | $2$ | $10$
$8$ | $8$ | $50$ | $3200$ | $1372$ | $4$ | $28$
$8$ | $15$ | $22$ | $2640$ | $378$ | $2$ | $18$
$8$ | $15$ | $43$ | $5160$ | $738$ | $2$ | $18$
$8$ | $36$ | $50$ | $14400$ | $2058$ | $2$ | $42$
$9$ | $5$ | $17$ | $765$ | $192$ | $3$ | $12$
$9$ | $5$ | $33$ | $1485$ | $372$ | $3$ | $12$
$9$ | $5$ | $49$ | $2205$ | $552$ | $3$ | $12$
$9$ | $9$ | $29$ | $2349$ | $588$ | $3$ | $21$
$9$ | $13$ | $41$ | $4797$ | $1200$ | $3$ | $30$
$10$ | $7$ | $37$ | $2590$ | $864$ | $4$ | $24$
$10$ | $10$ | $28$ | $2800$ | $312$ | $2$ | $12$
$10$ | $28$ | $37$ | $10360$ | $1152$ | $2$ | $32$
$11$ | $16$ | $46$ | $8096$ | $1620$ | $3$ | $36$
$12$ | $12$ | $34$ | $4896$ | $891$ | $3$ | $27$
$15$ | $8$ | $22$ | $2640$ | $378$ | $3$ | $18$
$15$ | $8$ | $43$ | $5160$ | $738$ | $3$ | $18$
$15$ | $8$ | $50$ | $6000$ | $1715$ | $5$ | $35$
$16$ | $6$ | $26$ | $2496$ | $500$ | $4$ | $20$
$16$ | $11$ | $46$ | $8096$ | $1620$ | $4$ | $36$
$25$ | $7$ | $37$ | $6475$ | $1080$ | $5$ | $30$
$28$ | $10$ | $37$ | $10360$ | $1152$ | $4$ | $32$
$36$ | $8$ | $50$ | $14400$ | $2058$ | $6$ | $42$

Table 1. Parameters for flag-transitive, $3$-chain-imprimitive $2$-designs
$\mathscr{D}^{\rm ft}(\mathbf{e};k)$ in Construction 4.2 with
$\mathbf{e}=(e_{1},e_{2},e_{3}),$ and $e_{1},e_{2},e_{3}\leq 50$

## References

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