Chain-imprimitive, flag-transitive 2-designs

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 $s \ge 2$, 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 $e_1e_2e_3$ points (where each $e_i\le 50$) 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 [1]. 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 $G \le {\text {Aut}}({\mathscr {D}})$ 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

$$\begin{aligned} \varvec{\mathscr {C}}: \ {\textstyle \left( {\begin{array}{c}{\mathscr {P}}\\ 1\end{array}}\right) } = {\mathscr {C}}_0 \prec {\mathscr {C}}_1 \prec \ldots \prec {\mathscr {C}}_{s-1} \prec {\mathscr {C}}_s = \{{\mathscr {P}}\} \end{aligned}$$

(1)

of partitions of ${\mathscr {P}}$. (Here $\left( {\begin{array}{c}{\mathscr {P}}\\ 1\end{array}}\right)$ 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 $G\le {\text {Aut}}({\mathscr {D}})$) 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 $s \ge 2$, there exist infinitely many 2-designs ${\mathscr {D}}=({\mathscr {P}},{\mathscr {B}})$ such that some subgroup $G\le {\text {Aut}}({\mathscr {D}})$ 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 $G\le {\text {Aut}}({\mathscr {D}})$ is flag-transitive on ${\mathscr {D}}$ and s-chain-imprimitive on ${\mathscr {P}}$, preserving the partitions in an s-chain $\varvec{\mathscr {C}}$ as in Eq. (1). Parameters of significance are the positive integers $e_1,\dots ,e_s$ such that, for $1 \le i \le s$, 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 $\varvec{\mathscr {C}}$ a chain of partitions as in Eq. (1), such that for $1 \le i \le s$ 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

$$\begin{aligned} y_i := 1 + \frac{k-1}{v-1} \left( \left( \prod _{j=1}^i e_j\right) - 1 \right) \quad \text{(so } \text{ in } \text{ particular } y_s = k\text{) }. \end{aligned}$$

(2)

Then there exists a 2-$(v,k,\lambda )$ design ${\mathscr {D}}$ with point set ${\mathscr {P}}$, for some $\lambda$, and a group $G\le {\text {Aut}}({\mathscr {D}})$ such that G is flag-transitive on ${\mathscr {D}}$ and preserves the partitions in the s-chain $\varvec{\mathscr {C}}$, 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 $\varvec{\mathscr {C}}$, 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 ${\mathscr {D}}'=({\mathscr {P}}, B^W)$ is also a 2-$(v,k,\lambda ')$ design with $W\le {\text {Aut}}({\mathscr {D}}')$ flag-transitive and preserving the same s-chain $\varvec{\mathscr {C}}$ of partitions of ${\mathscr {P}}$. So ${\mathscr {D}}'$ will have the same symmetry properties, the same point set ${\mathscr {P}}$ and partition chain $\varvec{\mathscr {C}}$, the same parameters $v,k, s, e_1,\dots ,e_s$, and possibly larger $\lambda '\ge \lambda$. 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 ${\mathscr {D}}'$. 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 Sect. 1.1 about the design constructions and some open questions. The organisation of the rest of paper is as follows: Sect. 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 Sect. 4 to justify our general construction method for flag-transitive s-chain-imprimitive 2-designs given in Construction 4.2. In Sect. 5 we prove Theorem 1.2, and in the final Sect. 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 $v=|{\mathscr {P}}|=e_1e_2$. The chain $\varvec{\mathscr {C}}$ in Eq. (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 ${\mathscr {D}}={\mathscr {D}}(e_1,e_2; \textbf{x}) = ({\mathscr {P}}, {\mathscr {B}})$ defined in [6, Sect. 2], where $\textbf{x}=(x_1,\dots , x_{e_2})$ with non-negative integer entries $x_1 \ge \ldots \ge x_{e_2}$, has as block set ${\mathscr {B}}$ the collection of all subsets $B\subseteq {\mathscr {P}}$ such that the sequence $\big (\, |B\cap C_i| \ | \ 1\le i\le e_2 \,\big )$ is a permutation of the sequence $\textbf{x}$. Thus $k:=|B|=\sum _i x_i$, and ${\text {Aut}}({\mathscr {D}})$ 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 ${\mathscr {D}}(e_1,e_2; \textbf{x})$ to be flag-transitive is that the tuple $\textbf{x}$ 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 Eq. (2). Since $\ell$ is a positive integer this equation implies that $v-1$ divides $(k-1)(e_1-1)$. Note that $\gcd (v-1, e_1-1)=\gcd (e_1e_2-1, e_1-1)=\gcd (e_2-1, e_1-1)$, 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 $k=1+(k-1)(e_1e_2-1)/(v-1)$ 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

$$\begin{aligned} z= \frac{(e_2-1)e_1}{d}\cdot \frac{k-1}{(v-1)/d}. \end{aligned}$$

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 ${\mathscr {D}}(e_1,e_2; \textbf{x})$, with $\textbf{x}=(\ell ^{k/\ell },0^{e_2-k/\ell })$ 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 1.1for the case $s\ge 3$.   For each $s \ge 3$ 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 $\varvec{\mathscr {C}}$ 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 $\varvec{\mathscr {C}}'$ of partitions ${\mathscr {C}}'_0, \ldots , {\mathscr {C}}'_{s-1}$ of ${\mathscr {P}}$ by deleting any nontrivial partition ${\mathscr {C}}_i$ ($1 \le i \le s-1$) from $\varvec{\mathscr {C}}$, so that ${\mathscr {C}}'_j = {\mathscr {C}}_j$ for $0 \le j \le i-1$ and ${\mathscr {C}}'_j = {\mathscr {C}}_{j+1}$ for $i \le j \le s-1$. Let $e'_j:= e_j$ for $0 \le j \le i-2$, $e'_{i-1}:= e_ie_{i+1}$, and $e'_j:= e_{j+1}$ for $i \le j \le s-1$. Then the partition ${\mathscr {C}}'_j$ contains $e'_j$ classes of ${\mathscr {C}}'_{j-1}$, for $1 \le j \le s-1$. 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 $\varvec{\mathscr {C}}$, where B is a k-subset of ${\mathscr {P}}$. Then $G \le W:= S_{e_1} \wr \ldots \wr S_{e_s}$, the stabiliser of the s-chain $\varvec{\mathscr {C}}$. 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 $\varvec{\mathscr {C}}'$ invariant, so W is a subgroup of the stabiliser $H:= S_{e'_1} \wr \ldots \wr S_{e'_{s-1}}$ of $\varvec{\mathscr {C}}'$. Define ${\mathscr {D}}':= \big ({\mathscr {P}}, B^H\big )$. Then ${\mathscr {D}}'$ 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 $\varvec{\mathscr {C}}'$.

(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 $e_1, e_2, e_3 \le 50$ 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. We give more details in Sect. 6.1. These parameter sets 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 $s\ge 3$. In particular, find more families with s unbounded.

(d) Comment on the automorphism groups of our examples.   We believe that the full automorphism group of each flag-transitive 2-design $\mathscr {D}^{\text {ft}}(\textbf{e},k)$ (where $\textbf{e}=(e_1,\dots ,e_s)$) in Construction 4.2 is equal to the iterated wreath product used in its construction. This is true if $s=2$ and we prove it also for $s=3$ (Lemma 6.4). We believe that this is true in general.

Problem 2

Prove that ${\text {Aut}}(\mathscr {D}^{\text {ft}}(\textbf{e};k))$ is equal to the wreath product $G= S_{e_1}\wr S_{e_2}\wr \dots \wr S_{e_s}$ for all 2-designs $\mathscr {D}^{\text {ft}}(\textbf{e};k)$ in Construction 4.2.

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 $\varvec{\mathscr {C}}$.

Given integers $s \ge 2$ and $e_1, \ldots , e_s \ge 2$. 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:

$$\begin{aligned} C_{(\delta _j)_{j>i}} := \big \{ (\varepsilon _j)_{j=1}^s \in {\mathscr {P}}\ \big | \ \varepsilon _j = \delta _j \ \text {for all} \ j > i \big \}. \end{aligned}$$

(3)

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 Eq. (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

$$\begin{aligned} {\mathscr {C}}_i := \Big \{ C_{(\delta _j)_{j>i}} \ \Big | \ (\delta _j)_{j>i} \in \prod _{j>i} \mathbb {Z}_{e_j} \Big \}. \end{aligned}$$

(4)

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 $\varvec{\mathscr {C}}$ as in Eq. (1). Moreover each set ${\mathscr {P}}$ and chain $\varvec{\mathscr {C}}$ 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

$$\begin{aligned} c_i = \prod _{j=0}^i e_j \quad \text {and} \quad d_i = \prod _{j=i+1}^s e_j. \end{aligned}$$

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

$$\begin{aligned} {\mathscr {C}}_i(X) := \{ C \in {\mathscr {C}}_i \ | \ X \cap C \ne \varnothing \}. \end{aligned}$$

(5)

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

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

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

$$\begin{aligned} {\mathscr {C}}_{i-1}(X) \cap {\mathscr {C}}_{i-1}(C)&= \{ C' \in {\mathscr {C}}_{i-1} \ | \ X \cap C' \ne \varnothing \ \text {and} \ C' \subseteq C \} \\&= \{ C' \in {\mathscr {C}}_{i-1} \ | \ C' \cap X \cap C \ne \varnothing \} \\&= {\mathscr {C}}_{i-1}(X \cap C). \end{aligned}$$

2.2 Iterated wreath product of groups

Using the notation of Sect. 2.1, for each $i \in \{1, \ldots , s\}$ let $G_i \le {\text {Sym}}(\mathbb {Z}_{e_i}) = S_{e_i}$. 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, Sect. 2.3], based on [3].

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

$$\begin{aligned} \varvec{\mathscr {C}}(C) : \ {\textstyle \left( {\begin{array}{c}C\\ 1\end{array}}\right) } = {\mathscr {C}}_0(C) \prec {\mathscr {C}}_1(C) \prec \ldots \prec {\mathscr {C}}_{j-1}(C) \prec {\mathscr {C}}_j(C) = \{C\} \end{aligned}$$

(6)

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

Proposition 2.1

Let $s \ge 2$, $1 \le i \le s$, and $C \in {\mathscr {C}}_i$, and let $G = S_{e_1} \wr \ldots \wr S_{e_s}$.

• (a) The stabiliser in ${\text {Sym}}(C)$ of the chain $\varvec{\mathscr {C}}(C)$ in Eq. (6) is $H(C):= G_C^C \cong S_{e_1} \wr \ldots \wr S_{e_i}$, and $H(C)^{{\mathscr {C}}_{i-1}(C)} \cong S_{e_i}$.

• (b) Further, for $1 \le j \le i-1$, $H(C) \cong \left( \prod _{C' \in {\mathscr {C}}_j(C)} H(C') \right) \rtimes H(C)^{{\mathscr {C}}_j(C)}$, where $H(C)^{{\mathscr {C}}_j(C)} \cong S_{e_{j+1}} \wr \ldots \wr S_{e_i}$ and $|{\mathscr {C}}_{j-1}(C)| = \prod _{\ell =j}^i e_\ell$.

• (c) In particular, for $1 \le j \le s-1$, $G = \left( \prod _{C' \in {\mathscr {C}}_j} H(C') \right) \rtimes G^{{\mathscr {C}}_j}$, 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 $H(C)_{({\mathscr {C}}_j(C))} = \prod _{C' \in {\mathscr {C}}_j} H(C') \cong (S_{e_1} \wr \ldots \wr S_{e_j})^{|{\mathscr {C}}_j(C)|}$, 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 $H(C) = G$. $\square$

2.3 Array of a subset

The array function of a nonempty subset B of ${\mathscr {P}}$, with respect to the chain $\varvec{\mathscr {C}}$ of partitions, is the function $\chi _B: \bigcup _{i=1}^s {\mathscr {C}}_i \rightarrow \mathbb {Z}_{\ge 0}$ defined by

$$\begin{aligned} \chi _B(C) := \left| B \cap C \right| \quad \text {for each }C \in {\mathscr {C}}_i\text { and each }i \in \{1, \ldots , s\}. \end{aligned}$$

(7)

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,

$$\begin{aligned} x_{(\delta _j)_{j>i}} := \chi _B\big (C_{(\delta _j)_{j>i}}\big ) \end{aligned}$$

(8)

for $C_{(\delta _j)_{j>i}}$ as in Eq. (3). For a subgroup $G \le S_{e_1} \wr \ldots \wr S_{e_s}$ which leaves invariant each partition in the chain $\varvec{\mathscr {C}}$, we say that two array functions $\chi _B$ and $\chi _{B'}$ are equivalent under G if $\chi _B^g = \chi _{B'}$ for some $g \in G$, where

$$\begin{aligned} \chi _B^g(C) := \chi _B\big (C^{g^{-1}}\big ) \quad \text {for all} \ C \in \bigcup _{i=1}^s {\mathscr {C}}_i. \end{aligned}$$

(9)

By [2, Lemma 2.5], if $G = S_{e_1} \wr \ldots \wr S_{e_s}$ then the G-orbit of $B \subseteq {\mathscr {P}}$ consists of all $B' \subseteq {\mathscr {P}}$ whose array function $\chi _{B'}$ 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 $\varvec{\mathscr {C}}$ of partitions ${\mathscr {C}}_i$ of ${\mathscr {P}}$ as in Eq. (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 $\varvec{\mathscr {C}}$ as in Eq. (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

$$\begin{aligned} \sum _{C \in {\mathscr {C}}_1} x_{C} \left( x_{C} - 1 \right) = \frac{k(k-1)}{v-1} (e_1 - 1) \end{aligned}$$

(10)

and

$$\begin{aligned} \text {for each }i \in \{2, \ldots , s\}, \quad \sum _{C \in {\mathscr {C}}_{i-1}} x_C \left( x_{C^+} - x_C \right) = \frac{k(k-1)}{v-1} (e_i - 1) \prod _{j \le i-1} e_j. \end{aligned}$$

(11)

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

$$\begin{aligned} \sum _{(\delta _j)_{j>i-1} \in \prod _{j>i-1} \mathbb {Z}_{e_j}} x_{(\delta _j)_{j>i-1}} \left( x_{(\delta _j)_{j>i}} - x_{(\delta _j)_{j>i-1}} \right) = \frac{k(k-1)}{v-1} (e_i - 1) \prod _{j \le i-1} e_j. \end{aligned}$$

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 $\varvec{\mathscr {C}}$ as in Eq. (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 $\varvec{\mathscr {C}}$, 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 $\varvec{\mathscr {C}}$.

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 $\varvec{\mathscr {C}}$ be a chain of nontrivial partitions of ${\mathscr {P}}$, as in Eq. (1) with parts as in Eq. (4), and chain stabiliser $G = S_{e_1} \wr \ldots \wr S_{e_s}$. For each $i \in \{0, \ldots , s\}$, let $y_i \in \mathbb {Z}^+$.

• (a) There exists a nonempty subset B of ${\mathscr {P}}$ which is uniform relative to $\varvec{\mathscr {C}}$ with uniform sequence $(y_0, \ldots , y_s)$, if and only if the following hold:

  $$\begin{aligned} \hbox {(i) }y_0 = 1, y_s=|B|, \quad \hbox { and }\quad \hbox { (ii) for all }i \in \{1, \ldots , s\}, y_{i-1} \mid y_i\hbox { with }\frac{y_i}{y_{i-1}} \le e_i. \end{aligned}$$

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

  $$\begin{aligned} B := \left\{ (\delta _i)_{i=1}^s \in {\mathscr {P}}\Big | 0 \le \delta _i \le \frac{y_i}{y_{i-1}} - 1 \ \text {for }\ 1 \le i \le s \right\} \end{aligned}$$

  (12)

  is uniform relative to $\varvec{\mathscr {C}}$ 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 $\varvec{\mathscr {C}}$ 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 $1\le i\le s$ 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 $C'\in {\mathscr {C}}_{i-1}(C)$, $x_{C'}\in \{0,y_{i-1}\}$, and $x_C=\sum _{C'\in {\mathscr {C}}_{i-1}(C)}x_{C'}$. It follows that $y_{i-1}\mid y_i$. Recall that $|{\mathscr {C}}_{i-1}(C)| = e_i$ so, if $C'\subset C$ is such that $x_{C'}=y_{i-1}$, then we have $x_C \le e_i x_{C'}$ and thus $\frac{y_i}{y_{i-1}} = \frac{x_C}{x_{C'}} \le e_i$, which is condition (ii). Therefore (i) and (ii) hold whenever some subset B is uniform relative to $\varvec{\mathscr {C}}$ with uniform sequence $(y_0, \ldots , y_s)$.

Conversely assume that conditions (i) and (ii) hold. Note that from condition (ii), $0 \le \frac{y_i}{y_{i-1}} - 1 < e_i$. 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 $\varvec{\mathscr {C}}$ 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

$$\begin{aligned} \chi _B\big (C_{(\delta _j)_{j>i}}\big ) = {\left\{ \begin{array}{ll} \prod _{\ell =1}^i \frac{y_\ell }{y_{\ell -1}} = y_i &{}\text {if }\delta _j < \frac{y_j}{y_{j-1}}\text { for all }j > i, \\ 0 &{}\text {otherwise}. \end{array}\right. } \end{aligned}$$

(13)

Therefore B is uniform relative to $\varvec{\mathscr {C}}$ 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 Eq. (12). Assume first that $B' \in {\mathscr {B}}$; so $B'=B^g$ for some $g\in G$. By [2, Lemma 2.5 (a)], the array function $\chi _{B'} = \chi ^g_B$ where $\chi ^g_B$ is as defined in Eq. (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 $\chi _{B'}(C)=\chi _{B}^g(C) = \chi _B\big (C^{g^{-1}}\big )$ which lies in $\{0,y_{i}\}$ since B has uniform sequence $(y_0, \ldots , y_s)$. Thus $B'$ is also uniform relative to $\varvec{\mathscr {C}}$ with uniform sequence $(y_0, \ldots , y_s)$.

Conversely, let $B'$ be a uniform subset of ${\mathscr {P}}$ relative to $\varvec{\mathscr {C}}$ with uniform sequence $(y_0, \ldots , y_s)$; we will show that $B' \in {\mathscr {B}}$.

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

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 Eq. (5), $|{\mathscr {C}}_{s-1}(B)|$ is the number of classes of ${\mathscr {C}}_{s-1}$ that intersect B nontrivially. Since B and $B'$ both have uniform sequence $(y_0, \ldots , y_s)$, it follows from Definition 3.1 and Lemma 3.2 that $|B|=|B'|=k=y_s$ and $|B\cap C|=y_{s-1}$ for $C\in {\mathscr {C}}_{s-1}(B)$, while $|B'\cap C|=y_{s-1}$ for $C\in {\mathscr {C}}_{s-1}(B')$. Thus $|{\mathscr {C}}_{s-1}(B)| = |{\mathscr {C}}_{s-1}(B')|= \frac{y_s}{y_{s-1}}$. 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 ${\mathscr {C}}_{s-1}(B')$. 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 ${\mathscr {C}}_{j}(B)^g = {\mathscr {C}}_{j}(B')$ where $j:= s-\ell +1$. Note that ${\mathscr {C}}_j(B)^g = {\mathscr {C}}_j(B^g)$ by Eq. (5). We proved above that $B^g$ is uniform relative to $\varvec{\mathscr {C}}$ with uniform sequence $(y_0, \ldots , y_s)$. Thus for any $C \in {\mathscr {C}}_j(B^g) = {\mathscr {C}}_j(B')$ we have $|{\mathscr {C}}_{j-1}(B^g \cap C)| = |{\mathscr {C}}_{j-1}(B' \cap C)| = \frac{y_j}{y_{j-1}}$. 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 ${\mathscr {C}}_{j-1}(B^g \cap C)^{h_C}={\mathscr {C}}_{j-1}(B' \cap C)$, 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 ${\mathscr {C}}_{j-1}(B')$. 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 ${\mathscr {C}}_0(B^g) = {\mathscr {C}}_0(B')$. Since ${\mathscr {C}}_0(B^g) = \left( {\begin{array}{c}B^g\\ 1\end{array}}\right)$ and ${\mathscr {C}}_0(B') = \left( {\begin{array}{c}B'\\ 1\end{array}}\right)$, it follows that $B^g = B'$. Therefore $B' \in {\mathscr {B}}$, which completes the proof of part (b). $\square$

Recall from Sect. 2.3 that the array functions $\chi _B$ and $\chi _{B'}$ of two subsets $B, B'\subseteq {\mathscr {P}}$ are equivalent under G if $\chi _B^g = \chi _{B'}$ 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 $B' \subseteq {\mathscr {P}}$ whose array function $\chi _{B'}$ 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 $\varvec{\mathscr {C}}$ be a chain of nontrivial partitions of ${\mathscr {P}}$, as in Eq. (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 $\varvec{\mathscr {C}}$ with uniform sequence $(y_0, \ldots , y_s)$. Then

$$\begin{aligned} |{\mathscr {B}}| = \prod _{j=1}^s \left( {\begin{array}{c}e_j\\ \frac{y_j}{y_{j-1}}\end{array}}\right) ^{k/y_j}. \end{aligned}$$

(14)

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 $C' \in {\mathscr {C}}_j(C)$, $\chi _{B \cap C}(C') = |B \cap C \cap C'| = |B \cap C'| = \chi _B(C') \in \{0,y_j\}$.

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 $i \ge 2$. 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 $b_i:= \prod _{j=1}^i \left( {\begin{array}{c}e_j\\ y_j/y_{j-1}\end{array}}\right) ^{y_i/y_j}$.

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 $\left( {\begin{array}{c}e_1\\ y_1\end{array}}\right)$. Therefore, recalling that $y_0 = 1$, we have

$$\begin{aligned} \big |(B \cap C_1)^{H(C_1)}\big | = \left( {\begin{array}{c}e_1\\ y_1\end{array}}\right) = \prod _{j=1}^1 \left( {\begin{array}{c}e_1\\ y_1/y_0\end{array}}\right) ^{y_1/y_1} = b_1, \end{aligned}$$

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

Suppose now that $i \ge 2$ 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 $B' \in (B \cap C_i)^{H(C_i)}$ has size $y_i$, and is uniform relative to ${\mathscr {C}}(C_i)$ with uniform sequence $(y_0, \ldots , y_i)$. The set $B'$ is the disjoint union of subsets $B' \cap C_{i-1}$ for all $C_{i-1} \in {\mathscr {C}}_{i-1}(B')$, where, by the observation at the start of this proof, each subset $B' \cap C_{i-1}$ has size $y_{i-1}$ and is uniform relative to ${\mathscr {C}}(C_i)$ with uniform sequence $(y_0, \ldots , y_{i-1})$. Since, by Claim 1, the set of all subsets of $C_{i-1}$ that are uniform relative to ${\mathscr {C}}(C_{i-1})$ with uniform sequence $(y_0, \ldots , y_{i-1})$ forms an orbit under the action of $H(C_{i-1})$, it follows that $B' \cap C_{i-1} \in (B \cap C_{i-1})^{H(C_{i-1})}$. Conversely, note that any subset $B''$ 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 $\big |\bigcup _{C_{i-1} \in {\mathscr {C}}_{i-1}(B \cap C_i)} B'' \big | = |B''| \cdot |{\mathscr {C}}_{i-1}(B \cap C_i)| = y_{i-1} \cdot y_i/y_{i-1}$. Thus, if we take one subset $B''$ 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 $B''$ 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

$$\begin{aligned} H(C_i) \cong \left( \prod _{C \in {\mathscr {C}}_{i-1}(C_i)} H(C) \right) \rtimes H(C_i)^{{\mathscr {C}}_{i-1}(C_i)} \cong \left( \prod _{C \in {\mathscr {C}}_{i-1}(C_i)} H(C) \right) \rtimes S_{e_i}. \end{aligned}$$

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

$$\begin{aligned} (B \cap C_i)^{H(C_i)} = \left( \bigcup _{C \in {\mathscr {C}}_{i-1}(B \cap C_i)} (B \cap C)^{H(C)} \right) ^{S_{e_i}} \end{aligned}$$

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

$$\begin{aligned} \big | (B \cap C_i)^{H(C_i)} \big | = \big | (B \cap C)^{H(C)} \big |^{y_i/y_{i-1}} \left( {\begin{array}{c}e_i\\ y_i/y_{i-1}\end{array}}\right) \quad \text {for any }C \in {\mathscr {C}}_{i-1}(B \cap C_i). \end{aligned}$$

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

$$\begin{aligned} \big | (B \cap C_i)^{H(C_i)} \big | = \Bigg (\prod _{j=1}^{i-1} \left( {\begin{array}{c}e_j\\ y_j/y_{j-1}\end{array}}\right) ^{y_{i-1}/y_{j}}\Bigg )^ {y_i/y_{i-1}} \cdot \left( {\begin{array}{c}e_i\\ y_i/y_{i-1}\end{array}}\right) = \prod _{j=1}^i \left( {\begin{array}{c}e_j\\ y_j/y_{j-1}\end{array}}\right) ^{y_i/y_j} = b_i. \end{aligned}$$

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 $|{\mathscr {B}}| = b_s = \prod _{j=1}^s \left( {\begin{array}{c}e_j\\ y_j/y_{j-1}\end{array}}\right) ^{k/y_j}$. This completes the proof of the lemma. $\square$

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 $\varvec{\mathscr {C}}$ 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 $\varvec{\mathscr {C}}$, the setwise stabiliser $G_{B}$ of B is transitive on B.

Proof

Let $B \subseteq {\mathscr {P}}$ be uniform relative to $\varvec{\mathscr {C}}$, 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 Eq. (12), it is enough to prove the result for such a subset B. Observe that the subset B in Eq. (12) can be written as $B = E_1 \times \ldots \times E_s$ where, for $i \in \{1, \ldots , s\}$,

$$\begin{aligned} E_i:= \left\{ \delta _i \in \mathbb {Z}_{e_i} \Big | \ 0 \le \delta _i < \frac{y_i}{y_{i-1}} \right\} = \mathbb {Z}_{y_i/y_{i-1}}. \end{aligned}$$

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 $\varvec{\mathscr {C}}^B: \left( {\begin{array}{c}B\\ 1\end{array}}\right) = {\mathscr {C}}_0^B \prec {\mathscr {C}}_1^B \prec \ldots \prec {\mathscr {C}}_s^B = \{B\}$. 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 ${\text {Sym}}(B)$ of this chain $\varvec{\mathscr {C}}^B$ is the group $L = {\text {Sym}}(E_1) \wr \ldots \wr {\text {Sym}}(E_s)$. Identify each group ${\text {Sym}}(E_i)$ with the subgroup of $S_{e_i}$ acting naturally on $E_i$ and fixing $\mathbb {Z}_{e_i} \setminus E_i$ pointwise; thus $L \le G$ and in particular $L \le G_B$. 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 $X \le {\text {Sym}}(\Gamma )$ and $Y \le {\text {Sym}}(\Delta )$ 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 $L = {\text {Sym}}(E_1) \wr {\text {Sym}}(E_2)$, so by [19, Lemma 5.4 (iii)] the group L is transitive on B. Suppose that $s \ge 3$ and that $K = {\text {Sym}}(E_1) \wr \ldots \wr {\text {Sym}}(E_{s-1})$ is transitive on $E_1 \times \ldots \times E_{s-1}$; then $L = K \wr {\text {Sym}}(E_s)$ 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. $\square$

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 $\varvec{\mathscr {C}}$ be a chain of nontrivial partitions of ${\mathscr {P}}$, as in Eq. (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 $\varvec{\mathscr {C}}$ 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. $\square$

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 Eq. (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 $1\le i\le s$ 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 $(k-1) \cdot d$ and, for $0\le i\le s$, let $c_i=\prod _{j=0}^ie_j$, with $e_0=1$.

• (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 $2\le i\le s$, $y_i-y_{i-1} = \frac{k-1}{v-1}(e_i-1){c_{i-1}}$.

• (b) If, in addition, for each $i \in \{1, \ldots , s\}$ the integer $y_{i-1}$ divides $(e_i-1){c_{i-1}}/d$, 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

(to delete)For $0\le i\le s$, let $c_i=\prod _{j=0}^ie_j$, with $e_0=1$. For each $i=1,\dots , s$,

$$\begin{aligned} c_i-1 = c_{i-1}(e_i-1) +\dots + c_{1}(e_{2}-1) + (e_1-1), \end{aligned}$$

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

$$\begin{aligned} y_i = 1 + \frac{u}{d} \cdot (c_i - 1) \quad \text {for }0 \le i \le s, \end{aligned}$$

(15)

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

$$\begin{aligned} y_i - y_{i-1} = \frac{u}{d} \cdot (c_i - c_{i-1}) = \frac{u}{d} \cdot (e_i - 1)c_{i-1}= \frac{k-1}{v-1}(e_i-1){c_{i-1}}. \end{aligned}$$

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 $\frac{(e_i - 1)c_{i-1}}{d}$. 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 $e_i\ge 2$ for all i, the $c_i$’s are strictly increasing. Then by Eq. (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 Eq. (15), we get that

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

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

$$\begin{aligned} e_i \left( 1 - \frac{u}{d} \right) + u \cdot \frac{c_{i}}{d} > \left( 1 - \frac{u}{d} \right) + u \cdot \frac{c_{i}}{d} = 1 + u \cdot \frac{c_i - 1}{d} = y_i. \end{aligned}$$

Therefore $\frac{y_i}{y_{i-1}} < e_i$. This proves part (b). $\square$

Now we give the general design construction.

Construction 4.2

Let $s \ge 2$ and $e_1, \ldots , e_s \ge 2$ 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 Eq. (2), that is,

$$\begin{aligned} y_i:= 1 + \frac{k-1}{v-1} \left( \left( \prod _{1\le j \le i} e_j \right) - 1 \right) \quad \text {for }i \in \{1, \ldots , s\}. \end{aligned}$$

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 $\varvec{\mathscr {C}}$ as in Eqs. (1) and (4). Also let $\textbf{e}:= (e_1, \ldots , e_s)$ and let $\mathscr {D}^{\text {ft}}(\textbf{e};k)=({\mathscr {P}},B^G)$, where B is as in Eq. (12), that is,

$$\begin{aligned} B = \bigg \{ (\delta _1, \ldots , \delta _s) \ \bigg | \ 0 \le \delta _i \le \frac{y_i}{y_{i-1}} - 1 \ \text {for }1 \le i \le s \bigg \}. \end{aligned}$$

Proposition 4.3

With the hypotheses of Construction 4.2,

• (a) The set B has cardinality k, and

• (b) $\mathscr {D}^{\text {ft}}(\textbf{e};k)$ 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 Eq. (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 $1< \frac{y_i}{y_{i-1}} < e_i$. 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, ${\mathscr {D}}= \mathscr {D}^{\text {ft}}(\textbf{e};k)$ 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 Eqs. (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 $i=1$,

$$\begin{aligned} \sum _{C \in {\mathscr {C}}_1} x_C (x_C - 1) = \sum _{C \in {\mathscr {C}}_1(B)} y_1 (y_1 - 1) = \left( \sum _{C \in {\mathscr {C}}_1(B)} y_1 \right) (y_1 - 1) = k(y_1 - 1). \end{aligned}$$

Condition Eq. (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

$$\begin{aligned} \sum _{C \in {\mathscr {C}}_{i-1}} x_{C} \left( x_{C^+} - x_{C} \right)= & {} \sum _{C \in {\mathscr {C}}_{i-1}(B)} y_{i-1} \left( y_i - y_{i-1} \right) \\= & {} \left( \sum _{C \in {\mathscr {C}}_{i-1}(B)} y_{i-1}\right) \cdot \left( y_i - y_{i-1} \right) = k(y_i - y_{i-1}). \end{aligned}$$

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 Eq. (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. $\square$

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 $\varvec{\mathscr {C}}$ as in Eqs. (1) and (4). So for $1 \le i \le s$ 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 Eq. (2) and that conditions Eqs. (10) and (11) of Theorem 2.2 hold. Then by Proposition 4.3, the design $\mathscr {D}^{\text {ft}}(\textbf{e};k)$ 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 $G\le W$ (so G leaves invariant the chain $\varvec{\mathscr {C}}$). 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 Eq. (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 $\varvec{\mathscr {C}}$ for some uniform sequence $(y'_0, \ldots , y'_s)$.

Claim 1: $(y'_0, \ldots , y'_s) = (y_0, \ldots , y_s)$.

By Eq. (2) and Lemma 3.2(a), we have $y'_0 = 1=y_0$ and $y'_s = k=y_s$. So assume that $1\le i<s$. Since B is the disjoint union of all the intersections $B \cap C$ with $C \in {\mathscr {C}}_i(B)$, we have $\sum _{C \in {\mathscr {C}}_i(B)} y'_i = \sum _{C \in {\mathscr {C}}_i(B)} |B \cap C| = |B| = k$. Also, for any $C \in {\mathscr {C}}_i$, we have $x_C = y'_i$ 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 Eqs. (10) and (11) in that theorem hold. We will show by induction that $y'_i=y_i$ for each $i\in \{1,\dots ,s\}$.

By Eq. (10) we have $\sum _{C \in {\mathscr {C}}_1} x_C (x_C - 1) = \frac{k(k-1)}{v-1} (e_1 - 1)$, while

$$\begin{aligned} \sum _{C \in {\mathscr {C}}_1} x_C (x_C - 1) = \sum _{C \in {\mathscr {C}}_1(B)} y'_1 (y'_1 - 1) = k(y'_1 - 1). \end{aligned}$$

Hence, $y'_1 = 1 + \frac{k-1}{v-1} (e_1 - 1)$ which equals $y_1$ by (2). Suppose now that $2\le i\le s$ and $y'_{i-1}=y_{i-1}$. 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 $C \in {\mathscr {C}}_{i-1}$, $x_C = y'_{i-1}$ and $x_{C^+} = y'_i$ whenever $C \in {\mathscr {C}}_{i-1}(B)$, and $x_C = 0$ otherwise, so

$$\begin{aligned} \sum _{C \in {\mathscr {C}}_{i-1}} x_C (x_{C^+} - x_C)= & {} \sum _{C \in {\mathscr {C}}_{i-1}(B)} y'_{i-1} (y'_i - y'_{i-1}) \\= & {} \left( \sum _{C \in {\mathscr {C}}_{i-1}(B)} y'_{i-1}\right) (y'_i - y'_{i-1}) = k(y'_i - y'_{i-1}). \end{aligned}$$

On the other hand, by Eq. (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

$$\begin{aligned} y'_i = y'_{i-1} + \frac{k-1}{v-1} (e_i - 1) \prod _{j \le i-1} e_j. \end{aligned}$$

(16)

By the induction hypothesis, we have

$$\begin{aligned} y'_i&= y_{i-1} + \frac{k-1}{v-1} (e_i - 1) \prod _{j \le i-1} e_j \\&= 1 + \frac{k-1}{v-1} \left( \Bigg (\prod _{j \le i-1} e_j\Bigg ) - 1 + (e_i - 1) \prod _{j \le i-1} e_j\right) \\&= 1 + \frac{k-1}{v-1} \left( \Bigg (\prod _{j \le i} e_j \Bigg ) - 1 \right) = y_i. \end{aligned}$$

Thus Claim 1 is proved.

Claim 2: Condition (FT1) of Theorem 1.2 holds.   Observe that in Eq. (16) (which holds for all $i\in \{1,\dots ,s\}$) the numbers $y'_i, y'_{i-1} \in \mathbb {Z}$, so also $\frac{k-1}{v-1} (e_i - 1) \prod _{j \le i-1} e_j \in \mathbb {Z}$. Now $v = \prod _{j=1}^s e_j$ so $\gcd \big ( v-1, \ \prod _{j \le i-1} e_j \big ) = 1$. 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

$$\begin{aligned} y_{i+1} - y_i = u \cdot \frac{e_{i+1} - 1}{d}\cdot \prod _{j \le i} e_j. \end{aligned}$$

Since B has uniform sequence $(y_0, \ldots , y_s)$ relative to $\varvec{\mathscr {C}}$ 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 $u \cdot \frac{e_{i+1} - 1}{d}\cdot \prod _{j \le i} e_j$, and since $y_i$ is coprime to u, $y_i$ divides $\frac{e_{i+1} - 1}{d}\cdot \prod _{j \le i} e_j$. 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 Eq. (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 $\varvec{\mathscr {C}}$ in Eq. (1), namely the design $\mathscr {D}^{\text {ft}}(\textbf{e}; k)$ 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 $\varvec{\mathscr {C}}$ 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 $\varvec{\mathscr {C}}$ with uniform sequence $(y_0, \ldots , y_s)$. In particular, ${\mathscr {D}}= \mathscr {D}^{\text {ft}}(\textbf{e};y_s) = \mathscr {D}^{\text {ft}}(\textbf{e};k)$ 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

$$\begin{aligned} e_1=d+1,\quad e_i = d + \prod _{j \le i-1} e_j, \quad \text {for }2\le i\le s,\text { and}\quad k:= 1 + \frac{v-1}{d} \text { where }v=\prod _{j \le s} e_j. \end{aligned}$$

(17)

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 $\varvec{\mathscr {C}}$ as in Eq. (1) with partitions as in Eq. (4). For $i\in \{1,\dots ,s\}$, let $y_i$ be as in Eq. (2), let B be as in Eq. (12), and let $\mathscr {D}^{\text {ft}}(\textbf{e};k)$ be as in Construction 4.2, where $\textbf{e} = (e_1, \ldots , e_s)$.

Proposition 6.2

The incidence structure $\mathscr {D}^{\text {ft}}(\textbf{e};k)$ 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 $b = \prod _{i=1}^s \left( {\begin{array}{c}e_i\\ y_{i}/y_{i-1}\end{array}}\right) ^{k/y_{i}}$ 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 $d':=\gcd (e_1-1,\dots ,e_s-1)$. We show inductively that d divides $e_i-1$ for each i: by Eq. (17) we have $d=e_1-1$ so this holds for $i=1$, and if $1<i\le s$ and d divides $e_j-1$ for all $j<i$, then by Eq. (17), $e_i\equiv d+1\equiv 1\pmod {d}$, so d also divides $e_i-1$. Thus d divides $d'$. On the other hand $d'\le e_1-1=d$ and hence $d=d'$.

Thus to prove that $\mathscr {D}^{\text {ft}}(\textbf{e};k)$ 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 Eq. (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 $1\le i<s$. Since $\frac{k-1}{v-1} = \frac{1}{d}$ we have, for each $i \in \{1, \ldots , s-1\}$, that $\prod _{j \le i} e_j = e_{i+1} - d$, so by Eq. (2),

$$\begin{aligned} y_i&= 1 + \frac{k-1}{v-1} \left( \left( \prod _{j \le i} e_j \right) - 1 \right) = 1 + \frac{1}{d} \left( \left( e_{i+1} - d \right) - 1 \right) \\&= \frac{d + (e_{i+1} - d - 1)}{d} = \frac{e_{i+1} - 1}{d}. \end{aligned}$$

This implies firstly that $y_i$ is a positive integer, since d divides $e_{i+1}-1$. Also it implies that $y_i$ divides $(e_{i+1} - 1)\left( \prod _{j \le i} e_j\right) /d$. Thus property (FT2) holds, and therefore $\mathscr {D}^{\text {ft}}(\textbf{e};k)$ 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 Eq. (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. $\square$

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

Proof of Theorem 1.1

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

6.1 Further examples

In this subsection let us consider the case where $s=3$. By Theorem 1.2, a flag-transitive, 3-chain-imprimitive 2-design $\mathscr {D}^{\text {ft}}(\textbf{e};k)$ exists if and only if the parameters $\textbf{e}=(e_1, e_2, e_3),$ and k satisfy the following conditions:

• (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) $y_1 = 1 + \frac{k-1}{e_1 e_2 e_3 - 1} (e_1 - 1)$ divides $\frac{(e_2 - 1)e_1}{d}$;

• (3) $y_2 = 1 + \frac{k-1}{e_1 e_2 e_3 - 1} (e_1 e_2 - 1)$ divides $\frac{(e_3 - 1)e_1 e_2}{d}$.

Moreover, such a design can be obtained by Construction 4.2. The input of this construction are the parameters $\textbf{e}$ and k, which are used to define a “generating” block

$$\begin{aligned} B = \left\{ (\delta _1,\delta _2,\delta _3) \;\; \Big |;\; 0 \le \delta _1 \le y_1, \;\; 0 \le \delta _2 \le \frac{y_2}{y_1} - 1, \;\; 0 \le \delta _3 \le \frac{k}{y_2} - 1 \right\} . \end{aligned}$$

The design $\mathscr {D}^{\text {ft}}(\textbf{e};k)$ has point set $\prod _{i=1}^3 \mathbb {Z}_{e_i}$ and block set $B^G$, where $G = S_{e_1} \wr S_{e_2} \wr S_{e_3}$. Moreover, by Proposition 4.3, $\mathscr {D}^{\text {ft}}(\textbf{e};k)$ is a 2-design admitting G as a flag-transitive, 3-chain-imprimitive group of automorphisms.

We implemented an exhaustive search using Magma [4] of all parameters $\textbf{e}$ and k satisfying the three arithmetic conditions above, for which $e_1, e_2, e_3 \le 50$. More precisely, our search considered each of the triples $(e_1, e_2, e_3)$, with $e_1, e_2, e_3 \le 50$, in lexicographic order. For each triple we determined $v=e_1 e_2 e_3$ and $d = \gcd (e_1 - 1, e_2 - 1, e_3 - 1)$. Then the integers $k<v$ for which condition (1) holds are precisely the integers of the form $k:= k_0x + 1$, where $k_0 = (v-1)/d$ and $x < d$ (note that d divides $v-1$). For each of these values of k, we then checked whether the divisibility conditions in (2) and (3) hold. The parameter sets for which all of these conditions hold are listed in Table 1, and moreover, as discussed in the previous paragraph, the parameter set in each line of Table 1 corresponds to a flag-transitive, 3-chain imprimitive 2-design $\mathscr {D}^{\text {ft}}(\textbf{e};k)$. We summarise these remarks in the following theorem.

Theorem 6.3

For $e_1, e_2, e_3 \le 50$ and $\textbf{e}:=(e_1, e_2, e_3),$ there exists $k\in \mathbb {Z}$ such that the conditions of Construction 4.2 are satisfied and $\mathscr {D}^{\text {ft}}(\textbf{e};k)$ is a 2-design admitting $S_{e_1} \wr S_{e_2} \wr S_{e_3}$ as a flag-transitive, 3-chain-imprimitive, group of automorphisms if and only if $(e_1, e_2, e_3, k)$ are as listed in Table 1.

Table 1 Parameters for flag-transitive, 3-chain-imprimitive 2-designs $\mathscr {D}^{\text {ft}}(\textbf{e};k)$ in Construction 4.2 with $\textbf{e}=(e_1, e_2, e_3),$ and $e_1, e_2, e_3 \le 50$

In Table 1 we indicate in boldface font those parameter sets which are obtained using the more explicit Construction 6.1. The fact that only three of the 57 parameter sets appear in bold suggests that there may be many more explicit constructions of infinite families of flag-transitive 3-chain-imprimitive 2-designs yet to be found (see Problem 1).

A natural question arises that we have been unable to resolve satisfactorily in general: namely whether, for a 2-design $\mathscr {D}^{\text {ft}}(\textbf{e};k)$ in Construction 4.2 with $\textbf{e}=(e_1, e_2,\dots , e_s)$, the iterated wreath product $G= S_{e_1}\wr S_{e_2}\wr \dots \wr S_{e_s}$ is equal to the full automorphism group of $\mathscr {D}^{\text {ft}}(\textbf{e};k)$. Since G contains a transposition, the only primitive subgroup containing G is the symmetric group $S_v$, where $v=e_1\dots e_s$, and $S_v$ does not preserve the set of blocks of $\mathscr {D}^{\text {ft}}(\textbf{e};k)$. Thus the full automorphism group $A = {\text {Aut}}(\mathscr {D}^{\text {ft}}(\textbf{e};k))$ is imprimitive on points. Since a wreath product $S_{a}\wr S_{b}$ is a maximal subgroup of $S_{ab}$, this means in particular that if $s=2$ then A is certainly equal to $G= S_{e_1}\wr S_{e_2}$. So the question concerns the designs with $s\ge 3$.

Further, the partitions ${\mathscr {C}}_i$ in the chain Eq. (1) are the only point partitions left invariant by G. Thus if $A\ne G$, then for at least one value of $i\in \{1,\dots ,s-1\}$, A must leave invariant the partition ${\mathscr {C}}_i$ of the chain Eq. (1). We sketch a proof that in the case $s=3$ we must have $A=G$. It would be interesting to know if this is the case for all s, as we believe it should be see Problem 2).

Lemma 6.4

Let $\mathscr {D}^{\text {ft}}(\textbf{e};k)$ be a 2-design given by Construction 4.2, where $\textbf{e}=(e_1, e_2,e_3)$. Then ${\text {Aut}}(\mathscr {D}^{\text {ft}}(\textbf{e};k)) = S_{e_1}\wr S_{e_2}\wr S_{e_3}$.

Proof

Suppose that $A\ne G$, where $A = {\text {Aut}}(\mathscr {D}^{\text {ft}}(\textbf{e};k))$ and $G = S_{e_1}\wr S_{e_2}\wr S_{e_3}$, and let $v=e_1e_2e_3$. Then, as discussed above, A is imprimitive on points, and A leaves invariant exactly one of the partitions ${\mathscr {C}}_1$ or ${\mathscr {C}}_2$ in the chain Eq. (1).

Suppose first that A leaves ${\mathscr {C}}_1$ invariant but not ${\mathscr {C}}_2$. Now the kernel K of the G-action on ${\mathscr {C}}_1$ is $K = (S_{e_1})^{e_2e_3}$, and this is the largest subgroup of $S_v$ fixing setwise each part of ${\mathscr {C}}_1$. Thus K is equal to the kernel of the A-action on ${\mathscr {C}}_1$, and as $A>G$, the induced group $A^{{\mathscr {C}}_1}$ must properly contain $G^{{\mathscr {C}}_1}= S_{e_2}\wr S_{e_3}$. Since $S_{e_2}\wr S_{e_3}$ is a maximal subgroup of $S_{e_2e_3}$, it follows that $A^{{\mathscr {C}}_1}= S_{e_2e_3}$, and hence $A=S_{e_1}\wr S_{e_2e_3}$. However this group does not leave invariant the block set of $\mathscr {D}^{\text {ft}}(\textbf{e};k)$. Indeed, consider the permutation g of ${\mathscr {P}}$ defined by

$$g:(\delta _1,\delta _2,\delta _3)\longrightarrow \left\{ \begin{array}{ll} (\delta _1,e_2,e_3)&{}\text { if }(\delta _2,\delta _3)=(0,0)\\ (\delta _1,0,0)&{}\text { if }(\delta _2,\delta _3)=(e_2,e_3)\\ (\delta _1,\delta _2,\delta _3)&{}\text { otherwise. } \\ \end{array}\right. $$

Then $g\in S_{e_1}\wr S_{e_2e_3}$ and swaps the ${\mathscr {C}}_1$-classes $C_{(0,0)}$ and $C_{(e_2,e_3)}$, which are as defined in Eq. (3). By its definition in Construction 4.2, the block B contains $y_1$ points from $C_{(0,0)}$ and no point from $C_{(e_2,e_3)}$ (since, by Lemma 4.1, $\frac{y_2}{y_1} < e_2$ and $\frac{y_3}{y_2} < e_3$). Thus the image $B^g$ contains $y_2 - y_1$ points from the ${\mathscr {C}}_2$-class $C_{(0)}$ and $y_1$ points from $C_{(e_3)}$. Hence $B^g$ is not uniform relative to $\varvec{\mathscr {C}}$ with uniform sequence $(y_0, \ldots , y_s)$, so by Lemma 3.2(b) the set $B^g$ does not belong in the orbit $B^G$. Therefore $B^g \notin {\mathscr {B}}$, and $S_{e_1}\wr S_{e_2e_3}$ does not preserve the block set ${\mathscr {B}}$, as claimed.

Hence A leaves ${\mathscr {C}}_2$ invariant but not ${\mathscr {C}}_1$. In this case, since the induced group $G^{{\mathscr {C}}_2}= S_{e_3}$, it follows that $A^{{\mathscr {C}}_2}= G^{{\mathscr {C}}_2}= S_{e_3}$, and hence the kernel K of the A-action on ${\mathscr {C}}_2$ properly contains the kernel $L= (S_{e_1}\wr S_{e_2})^{e_3}$ of the G-action on ${\mathscr {C}}_2$. It follows that, for some class $C\in {\mathscr {C}}_2$, the induced group $K^C$ properly contains the induced group $L^C = S_{e_1}\wr S_{e_2}$. The maximality of $S_{e_1}\wr S_{e_2}$ in $S_{e_1e_2}$ now implies that $K^C = S_{e_1e_2}$. From the transitivity of A on ${\mathscr {C}}_1$, we conclude that K is a subdirect subgroup of $(S_{e_1e_2})^{e_3}$, and then the fact that K contains $L= (S_{e_1}\wr S_{e_2})^{e_3}$ implies that $K= (S_{e_1e_2})^{e_3}$. Hence $A = S_{e_1e_2}\wr S_{e_3}$. Again, this group does not leave invariant the block set of $\mathscr {D}^{\text {ft}}(\textbf{e};k)$. Indeed, consider the permutation g of ${\mathscr {P}}$ that swaps the points (0, 0, 0) and $(0,e_2,0)$ of the ${\mathscr {C}}_2$-class $C_{(0)}$ and fixes all other points. Then $g\in A=S_{e_1e_2}\wr S_{e_3}$. By its definition in Construction 4.2, the block B contains $y_1$ points from $C_{(0,0)}$ and no point from $C_{(e_2,0)}$. In particular B contains (0, 0, 0) but not $(0,e_2,0)$. Thus the image $B^g$ contains $y_1-1$ points from $C_{(0,0)}$. Hence $B^g$ is not uniform relative to $\varvec{\mathscr {C}}$ with uniform sequence $(y_0, \ldots , y_s)$, and by Lemma 3.2(b) $B^g \notin {\mathscr {B}}$, and so $S_{e_1e_2}\wr S_{e_3}$ does not preserve the block set ${\mathscr {B}}$, as claimed. Thus we have a contradiction, and we conclude that G is the full automorphism group of $\mathscr {D}^{\text {ft}}(\textbf{e};k)$. $\square$