Groups Obtained from $2-(n,4,3)$ Supersimple Designs

TL;DR

This paper advances the classification of Conway groupoids linked to 2-(n,4,3) supersimple designs by improving bounds on hole stabilizers and providing partial classifications.

Contribution

It improves bounds for hole stabilizers to be primitive or contain the alternating group, aiding the classification of Conway groupoids for specific parameters.

Findings

Bounds for hole stabilizers are improved.

Partial classification achieved for lambda=3.

Enhanced understanding of group actions in design theory.

Abstract

We contribute towards the classification programme for Conway groupoids associated to a $2-(n,4,\lambda)$ design. Our main results improve the known bounds for a hole stabilizer to be primitive, or to contain the alternating group, ${\rm Alt}(n-1)$. We exploit these improved bounds to give a partial classification for Conway groupoids when $\lambda=3$.

Equations

$$\{\underline{\underline{(9,\mathrm{Alt}(4)\wr C_2)}},(13,?),(16,?),(17,?),(21,?)\}$$

$$\left\{\begin{array}[]{l}(13,M_{12}),(13,M_{11}),(16,\mathrm{SL}_{4}(2)),\underline{(16,\operatorname{Sym}(6))},\\ (16,\operatorname{Alt}(7)),(16,\operatorname{Alt}(6)),(17,?)\end{array}\right\}.$$

$$\mathcal{B}:=\{\{v_1,v_2,v_3,v_4\}:v_i\in \Omega \mathrm{and}{v}_1+v_2+v_3+v_4=0\}.$$

$${\Delta }_i^g={\Delta }_j.$$

$$[x,y]:=(x,y)\prod _{i=1}^{\lambda }(x_i,y_i),$$

$$\overline{a,b}:=\{x\in \Omega |\text{there exists ℓ∈B such that x,a,b,∈ℓ}\}.$$

$$[a_0,a_1,a_2,\dots ,a_n]=[a_0,a_1][a_1,a_2][a_2,a_3]\dots [a_{n-1},a_n]$$

$${\pi }_x(\mathcal{D}):=\{[a_0,a_1,\dots ,a_n]:a_0=a_n=x\}.$$

$$n⩽\frac{2t\lambda }{t-1}+1.$$

$${\Delta }_x\supseteq (\overline{\infty ,x}{)}^c.$$

$$\Omega \setminus \{\infty \}=\left({\Delta }_x\cup \overline{\infty ,x}\right)\setminus \{\infty \}.$$

$$n-1⩽\frac{n-1}{t}+2\lambda +1-1.$$

$${\Delta }_y={\Delta }_x^c\subseteq \overline{\infty ,x}.$$

$$\Lambda =\{(x_1,y)\mid x_1,y\in {\Delta }_x,x_1\ne y,y\in \overline{x_1,\infty }\}.$$

$${\Delta }_x\cdot (\text{Average number of points in Δx∩(x1,∞∖{x1})})⩾\frac{\lambda (n-1)}{6}\cdot 2$$

$$|{\Delta }_x\cap \overline{x,\infty }|⩾\frac{\lambda (n-1)}{3\Delta }+1,$$

$$n=|{\Delta }_x\cup \overline{\infty ,x}|⩽\Delta +2\lambda +2-\left(\frac{\lambda (n-1)}{3\Delta }+1\right).$$

$$n⩽\Delta +1+\frac{5\Delta \lambda }{3\Delta +\lambda }.$$

$$\frac{5\Delta \lambda }{3\Delta +\lambda }⩽\frac{10}{7}\lambda ,$$

$$B\supseteq \Omega \setminus \left(\overline{\infty ,a_1}\cup \overline{a_1,b}\cup \overline{\infty ,b}\right)\cup \{b\}.$$

$$[\infty ,a]=(b,c)(b_1,c_1)(b_2,c_2)\cdots (b_{\lambda -1},c_{\lambda -1});$$

$$[\infty ,b]=(a,c)(a_1,c_1^{\prime })(a_2,c_2^{\prime })\cdots (a_{\lambda -1},c_{\lambda -1}^{\prime });$$

$$\frac{2}{3}(n-1)⩽6\lambda -6.$$

$$n⩽\frac{\ell }{\ell -1}\left(6\lambda -7-\frac{1}{\ell }\right)⩽9\lambda -11,$$

$$n⩽\frac{6\ell }{\ell -1}\lambda +1⩽7.5\lambda +1,$$

$${\alpha }^{(b,g)}:\Delta \to \Gamma ,i\mapsto (i^{g^{-1}}\alpha {)}^{b_{i^{g^{-1}}}}.$$

$$(1\alpha )=(2\alpha {)}^{b_2}.$$

$$\frac{1}{3}d⩽6\lambda -6$$

$$d-d/\ell ⩽6\lambda +2$$

$$\begin{array}[]{rcl}\mathcal{D}&=&\{(1,2,3,4),(1,2,5,6),(1,2,7,8),(1,3,5,9),(1,3,6,7),(1,4,5,8),(1,4,7,9),\\[5.59721pt] &&(1,6,8,9),(2,3,5,7),(2,3,8,9),(2,4,5,9),(2,4,6,8),(2,6,7,9),(3,4,6,9),\\[5.59721pt] &&(3,4,7,8),(3,5,6,8),(4,5,6,7),(5,7,8,9)\}.\\[5.59721pt] \end{array}$$

$$\{\infty ,a,b_1,c_1\}\{\infty ,a,b_2,c_2\}\{\infty ,a,b_3,c_3\}.$$

Full text

1. Introduction

In his famous paper [Con97], John Conway used a “game” played on the projective plane ${\mathbb{P}}_{3}$ of order $3$ to construct the sporadic Mathieu group $M_{12}$, as well as a special subset of $\operatorname{Sym}(13)$ which he called $M_{13}$, and which could be endowed with the structure of a groupoid.

In recent work ([GGNS16, GGS18]), Conway’s construction has been generalized to geometries other than ${\mathbb{P}}_{3}$, namely to supersimple $2-(n,4,\lambda)$ designs. In this more general context, the analogue of the group $M_{12}$ is a subgroup of $\operatorname{Sym}(n-1)$ which is known as the hole stabilizer of the design. In this paper we prove a number of results concerning hole stabilizers. Our first main result is the following. It is a strengthening of [GGS18, Theorem E].

Theorem A**.**

Suppose that $\mathcal{D}$ is a supersimple $2-(n,4,\lambda)$ design, and that $\infty$ is a point in $\mathcal{D}$. Let $G:=\pi_{\infty}(\mathcal{D})$ be the hole stabilizer of $\infty$, considered as a permutation group via its natural embedding in $\operatorname{Sym}(n-1)$.

1. (1)

If $n>\frac{24}{7}\lambda+1$, then $G$ is transitive; 2. (2)

if $n>9\lambda-6$, then $G$ is primitive; 3. (3)

if $n\geqslant 10\lambda-5$ then $G$ is generously transitive; 4. (4)

if $n>9\lambda^{2}-12\lambda+5$, then one of the following holds:

1. (a)

$G$* contains $\operatorname{Alt}(n-1)$;* 2. (b)

$\lambda=1$, $\mathcal{D}=\mathbb{P}_{3}$ (the projective plane of order $3$), and $G=M_{12}$.

The value $\frac{24}{7}\lambda+1$ in the first bound in Theorem A is an improvement on the $4\lambda+1$ which appears in [GGS18, Theorem E]. In fact Lemmas 3.1 and 3.2, which are stated and proved in §3.1, give even stronger bounds, and the first bound in Theorem A follows directly from these results. The second bound in Theorem A is proved in §3.2.

The third bound in Theorem A is of a different flavour to results in [GGS18]. A definition is required: a permutation group $G\leqslant\operatorname{Sym}(n)$ is called generously transitive if for each $i,j\in\{1,2,\ldots,n\}$, $i\neq j$ exists an element $g\in G$ that interchanges this elements.

The fourth bound in Theorem A is already known and appears in [GGS18, Theorem E]. We keep it as part of our Theorem A as it will be useful later. However we can also give another result which has a similar flavour.

Theorem B**.**

Let $G=\pi_{\infty}(\mathcal{D})$. If $n>18\lambda-17$, then one of the following holds:

1. (1)

$\operatorname{Alt}(\Omega\setminus\{\infty\})\leqslant G$; 2. (2)

$\mathcal{D}$* is the projective plane of order $3$ and $G\cong M_{12}$;* 3. (3)

$G\cong\operatorname{Sym}(m)$* with $m\leqslant 3\lambda-1$, and the action of $G$ on $\Omega\setminus\{\infty\}$ is permutation isomorphic to the action on the set of $k$-subsets of $\{1,\dots,m\}$ for some $k\in\{2,\dots,\lfloor\frac{m}{2}\rfloor\}$;* 4. (4)

$G\cong\operatorname{Alt}(m)$* with $m\leqslant 2\lambda$, and the action of $G$ on $\Omega\setminus\{\infty\}$ is permutation isomorphic to the action on the set of $k$-subsets of $\{1,\dots,m\}$ for some $k\in\{2,\dots,\lfloor\frac{m}{2}\rfloor\}$.*

The strength of Theorem B is that it yields conclusions (1) and (2), conditional only a linear lower bound in $\lambda$ for $n$, as opposed to the quadratic lower bound in Theorem A (4).

On the other hand, the weakness of Theorem B is in conclusions (3) and (4): these parts have the advantage that they explicitly describe the permutation group $G$, however there is an associated loss of control on the size of $n$ in terms of $\lambda$. Nonetheless, for many values of $m$ and $k$ these actions violate the original quadratic bound that is the third bound in Theorem A and so, in principle, one could use this to obtain further restrictions on $m$ and $k$ in terms of $\lambda$. We have elected not to do this as it would introduce “clutter” to the statement, but practical applications of this theorem will probably require such an analysis. We prove Theorem B in §4.

Our final theorem extends the classification of hole stabilizers for small values of $\lambda$. When $\lambda=1$ or $2$, [GGNS16, Theorem C] gives a full classification. We now partially deal with the case of $\lambda=3$.

Theorem C**.**

Let $\mathcal{D}=(\Omega,\mathcal{B})$ be a $2-(n,4,3)$ supersimple design. Let $\infty\in\Omega$ and set $G:=\pi_{\infty}(\mathcal{D})$. Then either $G$ contains $\operatorname{Alt}(\Omega\setminus\{\infty\})$ or one of the following holds:

1. (1)

$G$* is intransitive: $(n,G)=\underline{\underline{(8,\{1\})}}$;* 2. (2)

$G$* is imprimitive: $(n,G)$ is in*

[TABLE] 3. (3)

$G$* is primitive $(n,G)$ is in*

[TABLE]

Furthermore, for those entries that are double underlined, all such examples are known and classified; for those entries that are single underlined, an example is known; for those entries that are not underlined, no such example is known.

Note that Theorem C asserts that the classification of hole stabilizers is complete for $\lambda=3$ except when $n\in\{13,16,17,21\}$. In §5 we prove Theorem C, and give a full description of all relevant examples.

One final remark: as we have said, our three main results fit into the programme of classification for Conway groupoids. In fact, though, we never directly study the groupoid of a design itself – all of our results are stated in terms of “the hole stabilizer”, and our proofs are also couched in these terms. For a definition of the Conway groupoid associated with a supersimple $2-(n,4,\lambda)$ design we refer to [GGS18] where, in addition, the connection between the hole stabilizer and the Conway groupoid is made clear.

1.1. Acknowledgments

It is a pleasure to thank B. McKay and M. Meringer who very kindly did a number of computer calculations at our request.

2. Background

2.1. Block designs

Let $t,n,k,\lambda$ positive integers. A balanced incomplete block design $(\Omega,\mathcal{B})$, also known as a $t-(n,k,\lambda)$ design, is a finite set $\Omega$ of size $n$, together with a finite multiset $\mathcal{B}$ each of size $k$ (called lines) such that any subset of $\Omega$ of size $t$ is contained in exactly $\lambda$ lines.

In this paper we are mostly interested in $2-(n,4,\lambda)$ designs. Such a design is called simple if there are no repeated lines, and supersimple if any two lines intersect in at most two points. In what follows we will be interested exclusively in supersimple $2-(n,4,\lambda)$ designs and so we can assume that the multiset $\mathcal{B}$ is in fact a set.

For some values of $t,n,k,\lambda$ the set of $t-(n,k,\lambda)$ designs have been completely enumerated. We will use this information for computer calculation purposes. We refer the reader to [CD07] for more information.

Let us note a particularly important example: the Boolean quadruple system of order $2^{k}$ is the design $(\Omega,\mathcal{B})$, where $\Omega$ is identified with the set of vectors of $\mathbb{F}_{2}^{k}$, and

[TABLE]

It is easy to see that $\mathcal{D}$ is a $2-(2^{k},4,2^{k-1}-1)$ design (in particular, when $k=2$, it is a $2-(8,4,3)$ design).

2.2. Permutation Groups

In this subsection, we collect some related notions about permutation groups that will be used at long of this paper. For more details we refer the reader to [DM96].

Suppose that $G$ is a group acting on a non-empty set $\Omega$. The action is called transitive if for all $x,y\in\Omega$ there is an element $g\in G$ such that $x^{g}=y$.

Suppose that the action of $G$ on $\Omega$ is transitive. A system of imprimitivity is a partition of $\Omega$ into $l$ subsets $\Delta_{1},\Delta_{2},\ldots,\Delta_{l}$ each of size $k$ such that $1<k,l<n$, and so that for all $i\in\{1,2,\ldots,l\}$ and all $g\in G$ there exists $j\in\{1,2,\ldots l\}$ such that

[TABLE]

The sets $\Delta_{i}$ are called blocks. We say that G acts imprimitively if there exists a system of imprimitivity. If no such set exists then G acts primitively on $\Omega$.

The support of an element $g\in G$, denoted $\operatorname{supp}(g)$ is the set of points in $\Omega$ not fixed by $g$.

2.3. Hole Stabilizers

Suppose that $\mathcal{D}=(\Omega,\mathcal{B})$ is a supersimple $2-(n,4,\lambda)$ design. Two points $x,y\in\Omega$ are collinear if there is some line in $\mathcal{B}$ that contains $x$ and $y$.

Suppose that a pair of distinct elements $x,y\in\Omega$ are collinear. We define the elementary move $[x,y]$ to be the permutation

[TABLE]

where $\{x,y,x_{i},y_{i}\}$ is a line in $\mathcal{B}$ for every $1\leqslant i\leqslant\lambda$. This product is well defined because $\mathcal{D}$ is supersimple. We also define $[x,x]=\mathrm{Id}_{\Omega}$.

Let $a$ and $b$ be distinct points in $\Omega$. We define

[TABLE]

In particular, note that $a,b\in\overline{a,b}$. Clearly the set of points in $\Omega$ moved by the permutation $[a,b]$ (also called the support of $[a,b]$) is precisely the set $\overline{a,b}$.

A move sequence is

[TABLE]

where $a_{i},a_{i+1}$ are collinear for all $0\leqslant i\leqslant n-1$. A move sequence is called closed if $a_{0}=a_{n}$. For each $x\in\Omega$ we define the hole stabilizer, $\pi_{x}(\mathcal{D})$, to be set of all closed move sequences such that $a_{0}=a_{n}=x$, that is

[TABLE]

It is easy to check that $\pi_{x}(\mathcal{D})$ is a subgroup of $\operatorname{Sym}(\Omega\setminus\{x\})=\operatorname{Sym}(n-1)$. In what follows we will need two easy facts [GGNS16, Lemma 3.1 and Theorem A].

Lemma 2.1**.**

Suppose that $\mathcal{D}=(\Omega,\mathcal{B})$ is a supersimple $2-(n,4,\lambda)$ design and that $x,y\in\Omega$.

1. (1)

$\pi_{x}(\mathcal{D})=\langle[x,a,b,x]:a,b\in\Omega\setminus\{x\}\rangle$. 2. (2)

$\pi_{x}(\mathcal{D})$* and $\pi_{y}(\mathcal{D})$ are conjugate subgroups of $\operatorname{Sym}(\Omega)$.*

The second statement above implies that all hole stabilizers for a supersimple design $\mathcal{D}$ are permutation isomorphic groups. This allows us to talk of “the” hole stabilizer $\mathrm{\mathbf{D}}$ (defined up to permutation isomorphism), and in the rest of this paper we denote this group as $\pi_{\infty}(\mathcal{D})$.

3. A proof of Theorem A

In this section we prove Theorem A. Throughout this section $\mathcal{D}$ is a supersimple $2-(n,4,\lambda)$ design with point set $\Omega$ and $\infty$ one such point. We write $G=\pi_{\infty}(\mathcal{D})$.

3.1. A bound for transitivity

The lemmas in this section immediately yield statement (1) in Theorem A.

Lemma 3.1**.**

Suppose that $G=\pi_{\infty}(\mathcal{D})$ has $t$ orbits on $\Omega\setminus\{\infty\}$ with $t>1$. Then

[TABLE]

In particular, if $n>4\lambda+1$, then $G=\pi_{\infty}(\mathcal{D})$ is transitive.

Proof.

Suppose that $x\in\Omega\setminus\{\infty\}$, and write $\Delta_{x}$ for the orbit of $x$ under $G$. Observe that

[TABLE]

Choose $x$ so that $\Delta_{x}$ is as small as possible. Then $|\Delta_{x}|\leqslant\frac{n-1}{t}$. Now, by the observation above,

[TABLE]

Noting that $x\in\Delta_{x}\cap\overline{\infty,x}$, we obtain that

[TABLE]

Rearranging this inequality gives the result. The “in particular” part of the lemma follows by taking $t=2$. ∎

Lemma 3.2**.**

Suppose that $G=\pi_{\infty}(\mathcal{D})$ has $2$ orbits on $\Omega\setminus\{\infty\}$. Then $n\leqslant\frac{24}{7}\lambda+1$.

Proof.

Note that if $\lambda\leqslant 2$, then [GGNS16, Theorem C] implies that $G=\pi_{\infty}(\mathcal{D})$ is always transitive. Thus we assume that $\lambda\geqslant 3$.

Suppose that $x\in\Omega\setminus\{\infty\}$. Write $\Delta_{x}$ for the orbit of $x$ under $G$, and note that (3.1) still holds. Suppose that $y\in\Omega\setminus(\Delta_{x}\cup\{\infty\})$. Now, taking complements of both sides of (3.1) we observe that

[TABLE]

Note, too, that (3.1) implies that $\Delta_{x}\cup\overline{\infty,x}=\Omega$. We wish to give a lower bound for $\Delta_{x}\cap\overline{\infty,x}$.

Observe that there are $\frac{\lambda(n-1)}{3}$ lines through $\infty$. All of these lines have either at least two elements of $\Delta_{x}$ or at least two of $\Delta_{y}$. Choose $x$ so that at least half of them (i.e. at least $\frac{\lambda(n-1)}{6}$ of them) contain at least two elements of $\Delta_{x}$.

Define

[TABLE]

Counting this in two different ways, we obtain that

[TABLE]

and so we conclude that there exists an element $x$ such that

[TABLE]

where $\Delta=|\Delta_{x}|$. This means that

[TABLE]

Rearranging we obtain that

[TABLE]

Now, for fixed $\Lambda$, the function $\frac{5\Delta\lambda}{3\Delta+\lambda}$ is an increasing function in the variable $\Delta$. Since $\Delta_{x}\subset\overline{\infty,y}$, we know that $\Delta\leqslant 2\lambda$ and we obtain that then

[TABLE]

and we obtain that $n\leqslant\Delta+1+\frac{10}{7}\lambda\leqslant\frac{24}{7}\lambda+1$. ∎

3.2. A bound for primitivity

In this section we prove statement (2) of Theorem A. Throughout this section we suppose that $G$ is transitive and preserves a system of imprimitivity with $\ell$ blocks each of size $k$ (so that $n-1=k\ell$). Let us start with the following lemma which is [GGNS16, Lemma 6.2].

Lemma 3.3**.**

If $n>4\lambda+1$, then at least one of the following holds:

1. (i)

if $a_{1},a_{2}\in\Omega$ lie in the same block of imprimitivity, then $\infty\in\overline{a_{1},a_{2}}$; 2. (ii)

$n\leqslant\frac{6\ell}{\ell-1}\lambda+1$.

Let us label blocks in the system of imprimitivity by $A,B,C,\dots$. Now we label points in $A$ by $a_{1},a_{2},a_{3},\dots$, points in $B$ by $b_{1},b_{2},b_{3},\dots,$ and so on.

Lemma 3.4**.**

Suppose that there exists a line $\{a_{1},a_{2},b,\infty\}$. Then $n\leqslant\frac{\ell}{\ell-1}(6\lambda-7-\frac{1}{\ell})$.

Proof.

Choose $x$, a point in $\Omega$ such that $x\not\in\overline{\infty,a_{1}}\cup\overline{a_{1},b}\cup\overline{\infty,b}$. Let $g_{x}=[\infty,a_{1},x,\infty]$ and observe, first, that $a_{2}^{g_{x}}=b$. Thus $A^{g_{x}}=B$. Observe, second, that $a_{1}^{g_{x}}=x$ and so $x\in B$. We conclude that

[TABLE]

In particular, $|B|\geqslant n-(6\lambda-6)+1$. Now use the fact that $|B|=\frac{n-1}{\ell}$, and the result follows. ∎

Lemma 3.5**.**

Suppose that $G$ preserves a system of imprimitivity with $\ell=2$ blocks of size $\frac{n-1}{2}$. Then $n\leqslant 6\lambda+3$.

Proof.

This implies that $G$ contains an element of support of size $2k=n-1$ in any generating set. Now the result follows from the fact that $G$ is generated by elements with support of size at most $6\lambda+2$ ([GGNS16, Lemma 7.3] – or see item (4) of Lemma 2.1).

∎

The following lemma is stated for $\ell=3$; it is possible that similar statements may hold more generally.

Lemma 3.6**.**

Suppose that $\ell=3$ and that any line containing $\infty$ contains points from all blocks of imprimitivity ($A$, $B$ and $C$). Then $n\leqslant 9\lambda-8$.

Proof.

Let $L=\{\infty,a,b,c\}$ be a line. Then observe that

[TABLE]

Now consider the element $g=[\infty,a,b,\infty]$. If $g$ is to fix $B$ set-wise, then $[a,b]$ must move $c,c_{1},\dots,c_{\lambda-1}$ to elements in $B$. If this is the case, then $g$ must interchange $A$ and $C$. The same argument works if we consider what happen when we fix $A$ or $C$ set-wise.

We conclude that in any case $g$, which is an element of support at most $6\lambda-6$, must move at least two blocks, and so

[TABLE]

∎

Lemma 3.7**.**

Suppose that $L$ is any line containing $\infty$, then $L$ intersects a block of imprimitivity in at most $1$ point. Then a block of imprimitivity has size at most $2\lambda-1$.

Proof.

Let $L=\{\infty,a,b,c\}$ and suppose that $x\in B$, $x\not\in\overline{a,\infty}$ and $x\not\in\overline{a,b}$. Observe that the supposition implies that $x\not\in\overline{b,\infty}$. Now let $g_{x}=[\infty,a,x,\infty]$ and observe that $a^{g_{x}}=x\in B$ and $c^{g_{x}}=b\in B$. This is a contradiction.

Thus either $x\in\overline{a,\infty}$ or $x\in\overline{a,b}$. The supposition ensures that $|B\cap\overline{a,\infty}|\leqslant\lambda$. Suppose, then that $x\in\overline{a,b}$ and $x\not\in\overline{a,\infty}$. Then there is a line $\{a,b,x,y\}$ and, defining $g_{x}$ as before, observe that $a^{g_{x}}=x\in B$ and $c^{g_{x}}=y^{[x,\infty]}$. If $y\in B$, then the supposition guarantees that $c^{g_{x}}=y^{[x,\infty]}=y\in B$, which is a contradiction. We conclude that $y\not\in B$. Thus $\overline{a,b}$ can contain at most $\lambda-1$ points of $B$ apart from $b$. The result follows. ∎

Let us sum up the work of this section with the next lemma which is statement (2) of Theorem A.

Lemma 3.8**.**

If $G$ preserves a non-trivial system of imprimitivity, then $n\leqslant 9\lambda-6$.

Proof.

If $\lambda\leqslant 2$, then the result follows immediately from [GGNS16, Theorem C]. Assume from here on that $\lambda\geqslant 3$. If $\ell=2$, then Lemma 3.5 implies that $n\leqslant 6\lambda+3$ and the result follows.

Suppose from here on that $\ell\geqslant 3$. If there exists a line $\{a_{1},a_{2},b,\infty\}$, then Lemma 3.4 implies that

[TABLE]

and the result follows.

Suppose from here on that if $L$ is any line containing $\infty$, then $L$ intersects a block of imprimitivity in at most $1$ point. If $\ell=3$, then Lemma 3.6 implies that $n\leqslant 9\lambda-8$, and the result follows. If $\ell=4$, then Lemma 3.7 implies that $n\leqslant 8\lambda-3$, and the result follows.

Suppose from here on that $\ell\geqslant 5$. Then Lemma 3.3 implies that

[TABLE]

and the result follows for $\lambda\geqslant 4$. For $\lambda=3$, (3.2) implies that $n\leqslant 23$. Since $2-(n,4,3)$ designs only occur for $n\equiv 0,1\pmod{4}$, we conclude that $n\leqslant 21$, and the result follows. ∎

3.3. A bound for generous transitivity

In this section we prove the third bound in Theorem A.

Lemma 3.9**.**

Let $G=\pi_{\infty}(\mathcal{D})$. If $n\geqslant 10\lambda-5$ then $G$ is generously transitive.

Proof.

Let $a,b\in\Omega\setminus\{\infty\}$. We must find $g\in G$ such that $a^{g}=b$. If $\infty\not\in\overline{a,b}$ then we can use $g:=\left[\infty,a,b,\infty\right]$.

Suppose that $\infty\in\overline{a,b}$. This means that there exists a line $\{\infty,a,b,c\}$ for some $c$. Choose $x$ such that $x$ is not in $\overline{a,b}$, $\overline{\infty,a}$, $\overline{\infty,b}$, $\overline{a,c}$, $\overline{b,c}$. Then we can take $g$ to be $\left[\infty,c,x,\infty\right]$. Finally, observe that the sets listed above together contain at most $10\lambda-6$ elements, so, assuming $n\geqslant 10\lambda-5$ we obtain the result.

∎

4. A proof of Theorem B

Our aim in this section is to prove Theorem B. We need two background results. The first is [GGS18, Theorem D].

Theorem 4.1**.**

Suppose that $\mathcal{D}$ is a supersimple $2-(n,4,\lambda)$ design, and that $[\infty,a,b,\infty]=(\,)$ whenever $\infty$ is collinear with $a,b$. Then one of the following is true:

1. (1)

$\mathcal{D}$* is a Boolean quadruple system and $\pi_{\infty}(\mathcal{D})$ is trivial;* 2. (2)

$\mathcal{D}$* is the projective plane of order $3$ and $\pi_{\infty}(\mathcal{D})\cong M_{12}$; or* 3. (3)

$\pi_{\infty}(\mathcal{D})\supseteq\operatorname{Alt}(\Omega\setminus\{\infty\})$.

We also need a result of Liebeck and Saxl [LS91, Theorem 2].

Theorem 4.2**.**

Let $G$ be a primitive group of degree $d$. Then either

1. (1)

all non-trivial elements have support at least $\frac{1}{3}d$ points, or 2. (2)

$G$* is a subgroup of $\operatorname{Sym}(m)\wr\operatorname{Sym}(r)$ containing $(\operatorname{Alt}(m))^{r}$, with $m\geqslant 5$, where the action of $\operatorname{Sym}(m)$ is on $k$-element subsets of $\{1,\dots,\ell\}$ and the wreath product has the product action of degree $d=\binom{\ell}{k}^{r}$.*

We note that there is an improvement on Liebeck and Saxl’s result due to Guralnick and Magaard [GM98] – we have elected not to use their result as it includes a longer list of exceptions.

Recall that the product action of $\operatorname{Sym}(\ell)\wr\operatorname{Sym}(r)$ can be thought of as an action on the set of functions $\Delta\to\varGamma$, where $\Delta$ is a set of size $r$ and $\varGamma$ is a set of size $\ell$. Let $bg=(b_{1},\dots,b_{r})g$ be an element of $\operatorname{Sym}(\ell)\wr\operatorname{Sym}(r)$ (so $b_{1},\dots,b_{r}\in\operatorname{Sym}(\ell)$ and $g\in\operatorname{Sym}(r)$), then for $\alpha:\Delta\to\varGamma$, we have

[TABLE]

Note that there are $d=\ell^{r}$ functions $\Delta\to\varGamma$. Using the notation just established, we have the following lemma.

Lemma 4.3**.**

1. (1)

Let $G=\operatorname{Sym}(\ell)\wr\operatorname{Sym}(r)$, considered as a permutation group via the product action on $d=\ell^{r}$ points. Suppose that $g=bh\in G$, with $b\in\operatorname{Sym}(\ell)^{r}$, $h\in\operatorname{Sym}(r)$ and $h\neq 1$. Then the number of fixed points of $bh$ is maximal when $h$ is a transposition, and $b=1$. In this case $bh$ fixes $\ell^{r-1}=d/\ell$ points. 2. (2)

Let $G=\operatorname{Sym}(m)$ acting on the set $\Lambda$ of $k$-subsets of $\{1,\dots,m\}$ for some $k\in\{2,\dots,\lfloor\frac{m}{2}\rfloor\}$.

1. (a)

If $g\in\operatorname{Sym}(m)\setminus\{1\}$, then $g$ moves at least $2m-4$ points of $\Lambda$. 2. (b)

If $g\in\operatorname{Alt}(m)\setminus\{1\}$, then $g$ moves at least $3m-6$ points of $\Lambda$.

Proof.

For (1), let $h$ be non-trivial, and label elements so that $1^{h}=2$. If $g=bh$ fixes a function $\alpha$, then we require that

[TABLE]

Thus the image of $1$ under $\alpha$ is prescribed by the image of $2$, and we obtain immediately that there are at most $\ell^{r-1}$ possibilities for $\alpha$.

For (2), let $g$ be non-trivial, and label elements so that $1^{g}=2$. The number of $k$-sets that contain $1$ but don’t contain $2$ is $\binom{m-2}{k-1}$; likewise the number of $k$-sets that contain $2$ but don’t contain $2^{g}$ is $\binom{m-2}{k-1}$. These two families of sets are disjoint, and all sets contained therein are moved by $g$, hence $2\binom{m-2}{k-1}$ is a lower bound on the number of points moved by $g$.

If $k>2$ and $m>5$, then this immediately yields the lower bound $3m-6$ (recall that we may assume that $k\leqslant m/2$). Thus we must consider the cases $m\leqslant 5$ or $k=2$; note, though, that if $m\leqslant 5$, then we automatically have that $k\leqslant 2$.

Suppose, then, that $k=2$. If in the cycle decomposition of $g$, we have $(1,2,\dots,t)$, then the number of $2$-sets containing $1$ but not $2$, then $2$ but not $3$ (and so on ) is at least $t(m-2)$. Thus if $g$ contains a cycle of length $3$ or more, then the result follows; the bound for (a) also follows. Suppose, then that $g$ is in $\operatorname{Alt}(m)$ and $g$ is a product of $k$ distinct transpositions with $k\geqslant 2$; write $g=(1,2)(3,4)\cdots$. Then the same argument yields a lower bound of $4m-8$, and the result follows. ∎

We are ready to prove Theorem B.

Proof of Theorem B.

The result is true for $\lambda\leqslant 2$ by classification theorems in [GGNS16]. Note that $n>18\lambda-17>9\lambda+1$ for $\lambda\geqslant 3$ and so $G=\pi_{\infty}(\mathcal{D})$ is a primitive subgroup of $\operatorname{Sym}(n-1)$.

Now, by Theorem 4.1, we can assume that $[\infty,a,b,\infty]\neq 1$ for some $a,b$ collinear with $\infty$. Such an element has support at most $6\lambda-6$. Now we consider the possibilities given in Theorem 4.2. If possibility (1) occurs, then we conclude that $n=d-1$ with

[TABLE]

which is a contradiction.

Thus, possibility (2) occurs: $G$ is a subgroup of $\operatorname{Sym}(\ell)\wr\operatorname{Sym}(r)$ in the product action on $\ell^{r}$ points. If $r\neq 1$, then any set of generators for $G$ must include an element $bg$ with $g\neq 1$ (using the notation established before Lemma 4.3). However we know that the set of elements of the form $[\infty,a,b,\infty]$ generate $\pi_{\infty}(\mathcal{D})$ and these elements have support at most $6\lambda+2$. Referring to Lemma 4.3, we conclude that $n=d-1$ with

[TABLE]

and so $d<12\lambda+4$ which is a contradiction for $\lambda\geqslant 4$. For $\lambda=3$, we have a contradiction when $\ell\neq 2$. When $\ell=2$ we must rule out $n\in\{37,38,39,40\}$ but, since none of these are powers of $2$, this is immediate.

Thus we are left with the possibility that $r=1$, $d=\binom{m}{k}$, and $G=\pi_{\infty}(\mathcal{D})$ is either $\operatorname{Sym}(m)$ or $\operatorname{Alt}(m)$ with the action on $\Omega\setminus\{\infty\}$ isomorphic to the action on the set $\Lambda$ of $k$-subsets of $\{1,\dots,m\}$.

If $G\cong\operatorname{Sym}(m)$, then Lemma 4.3 implies that a non-trivial element of $G$ must move at least $2m-4$ points of $\Lambda$. We know that there exist non-trivial elements that move at most $6\lambda-6$ elements, and so we conclude that $m\leqslant 3\lambda-1$.

If $G\cong\operatorname{Alt}(m)$ with $m>5$, then Lemma 4.3 implies that a non-trivial element of $G$ must move at least $3m-6$ points of $\Lambda$, and the same argument implies that $m\leqslant 2\lambda$.

If $G\cong\operatorname{Alt}(5)$, then Lemma 4.3 implies that a non-trivial element of $G$ must fix at least $8$ points of $\Lambda$. We conclude that $6\lambda-6\geqslant 8$ and so $\lambda\geqslant 3$, and we are done. ∎

5. Theorem C

Our aim in this section is to classify puzzle groups arising from supersimple $2-(n,4,3)$ designs. Note, first, that such designs only occur for $n\equiv 0,1\pmod{4}$ and $n\geqslant 8$.

Throughout this section we let $\mathcal{D}$ be a supersimple $2-(n,4,3)$ design and set $G=\pi_{\infty}(\mathcal{D})$. Since $\lambda=3$, we observe that all elementary moves are even permutations and so, by Lemma 2.1, $G$ is a subgroup of $\operatorname{Alt}(n-1)$. We start by applying Theorem A to this situation in which case we obtain the following lemma.

Lemma 5.1**.**

1. (1)

if $n>11$, then $G$ is transitive; 2. (2)

if $n>21$, then $G$ is primitive; 3. (3)

if $n>50$, then $G=\operatorname{Alt}(n-1)$.

5.1. Small $n$

The $2-(8,4,3)$ and $2-(9,4,3)$ designs are listed explicitly in [CD07]. Direct calculation then yields the following result.

Lemma 5.2**.**

The following statements holds:

1. (1)

There is a unique supersimple $2-(8,4,3)$ design, and its hole stabilizer is trivial. 2. (2)

There is a unique supersimple $2-(9,4,3)$ design, and its hole stabilizer, $G$, is transitive and imprimitive, with $G\cong\operatorname{Alt}(4)\wr C_{2}$.

Proof.

Using the list in [CD07], for $n=8$, we can see that there exists exactly one $2-(8,4,3)$ supersimple design. This designs is (isomorphic to) the Boolean quadruple system of order $8$, and so $\pi_{\infty}(\mathcal{D})$ is trivial.

For $n=9$ we can also check that exists exactly one $2-(9,4,3)$ supersimple design. A calculation using [GAP19] shows that $\pi_{\infty}(\mathcal{D})\cong\operatorname{Alt}(4)\wr C_{2}$. ∎

Let us be explicit for the case $n=9$: it turns out that the only supersimple $2-(9,4,3)$ design is

[TABLE]

Next, a computer calculation of Professor Brendan McKay confirms that there are 28,893 supersimple $2-(12,4,3)$ designs; more computer calculations with [GAP19] confirm that all of these designs have hole stabilizer isomorphic to $\operatorname{Alt}(11)$, thus we assume that $13\leqslant n\leqslant 29$ from here on.

From here on we assume that $n\geqslant 13$. Lemma 5.1 implies, then, that $G$ is transitive.

5.2. The imprimitive case

Suppose that $G$ is transitive and preserves a system of imprimitivity with $\ell$ blocks of size $k$ (so $n-1=k\ell$). Lemma 5.1 implies that $n\leqslant 21$. We know that $n-1$ cannot be prime, so this implies that $n\in\{13,16,17,21\}$.

5.3. The primitive case

In this section we assume that $G$ is primitive and not isomorphic to $\operatorname{Alt}(n-1)$. We know already, thanks to Lemma 5.1, that $n\leqslant 50$ and, thanks to Lemma 5.2, that $n\geqslant 13$. We start by improving this.

Lemma 5.3**.**

Suppose that $G$ is primitive. Then either $G\cong\operatorname{Alt}(n-1)$ or one of the following statements holds:

• •

$n=13$* and $G\in\{M_{12},M_{11}\}$;*

• •

$n=16$* and $G\in\{\mathrm{SL}_{4}(2),\operatorname{Sym}(6),\operatorname{Alt}(7),\operatorname{Alt}(6)\}$;*

• •

$n=17$* and $G$ is isomorphic to one of $18$ primitive groups in $2^{4}.\mathrm{SL}_{4}(2)$;*

• •

$n=28$* and $G\in\{\mathrm{PSp}_{4}(3)\rtimes C_{2}\}$;*

• •

$n=29$* and $G\in\{\mathrm{Sp}_{6}(2),\operatorname{Sym}(8)\}$.*

Proof.

We know, by Theorem 4.1, that there exist points $a,b\in\Omega$ such that $g=[\infty,a,b,\infty]$ is non-trivial and $a,b$ are collinear with $\infty$. Then $g$ is an element with support of size at most $6\lambda-6=12$.

Now the list above contains all but one of the primitive groups on $n-1$ points which

1. (1)

satisfy $13\leqslant n\leqslant 50$ with $n\equiv 0,1\pmod{4}$; 2. (2)

contain a non-trivial element with support at most $12$; 3. (3)

are subgroups of $\operatorname{Alt}(n-1)$.

Let us consider the missing entry which occurs when $n=13$ and $G\cong\operatorname{PSL}_{2}(11)$. It is easy to check that there are no non-trivial elements that fix more than 2 points. But now, by Theorem 4.1, we can assume that there exists $g:=[\infty,a,b,\infty]$ which is not trivial and for which there exists $c$ such that $\{\infty,a,b,c\}\in\mathcal{B}$. But now observe that $g$ fixes $a$, $b$ and $c$, and so we have a contradiction. ∎

Lemma 5.4**.**

$n\leqslant 17$.

Proof.

Suppose that $n>17$. Then $n\in\{28,29\}$. The three possible permutation groups given in Theorem 5.3 have precisely one non-trivial conjugacy class of elements of support at most $12$. In every case it is a conjugacy class of involutions with support exactly $12$.

Using [GAP19] one can verify that if $G$ is one of these three permutation groups, $g,h\in G$ are two involutions of support $12$ and $\tau$ is one of the six disjoint transpositions whose product is $g$, then $\tau$ is not one of the six disjoint transpositions whose product is $h$.

Let $a\in\Omega\setminus\{\infty\}$ and consider the three lines connecting $\infty$ to $a$:

[TABLE]

Consider the permutation $[\infty,a,b_{1},\infty]$. This is either trivial, or has support $12$. Note that in the latter case, this implies that the sets $\overline{\infty,a}$, $\overline{a,b_{1}}$ and $\overline{b_{1},\infty}$ must overlap only in the set $\{\infty,a,b_{1},c_{1}\}.$

On the other hand if $[\infty,a,b_{1},\infty]$ is trivial, then one can check that one must have $b_{2}\in\overline{a,b_{1}}$ and one obtains that $[\infty,a,b_{2},\infty]$ is trivial, likewise $[\infty,a,b_{3},\infty]$.

By running the same argument starting with $b_{2}$ and $b_{3}$ in place of $b_{1}$, one sees that $[\infty,a,b_{1},\infty]$ is trivial if and only if $[\infty,a,b_{2},\infty]$ is trivial if and only if $[\infty,a,b_{3},\infty]$ is trivial.

Now Theorem 4.1 implies that we can choose $a$ so that $[\infty,a,b_{1},\infty]$ is not trivial. Thus the same is true of $[\infty,a,b_{2},\infty]$. But now, note that both $[\infty,a,b_{1},\infty]$ and $[\infty,a,b_{2},\infty]$ include the transposition $(b_{3},c_{3})$. On the other hand $[\infty,a,b_{1},\infty]$ includes the transposition $(b_{2},c_{2})$ which $[\infty,a,b_{2},\infty]$ does not. This is a contradiction and we are done. ∎

Lemma 5.4 completes the proof of Theorem C. The remaining couple of results rule out some of the open possibilities from Theorem C. The first of these results generalizes the idea of Lemma 5.4.

Lemma 5.5**.**

Let $G:=\pi_{\infty}(\mathcal{D})$ a puzzle group, where $\mathcal{D}:=(\Omega,\mathcal{B})$ is a $2-(n,4,\lambda)$ supersimple design. Then, one of the next statements is true:

1. (1)

there exists a non trivial element of support strictly less than $6(\lambda-1)$. 2. (2)

there exist two different elements $g,h$, both with cycle type $2^{3(\lambda-1)}$ and so that in their cycle decomposition they have a common transposition.

Proof.

Suppose that (1) is not true, i.e. suppose that the unique element of $G$ with support less than $6(\lambda-1)$ is the identity. Let $a\in\Omega\smallsetminus\{\infty\}$, and consider the $\lambda$ lines connecting $\infty$ to $a$:

[TABLE]

Consider the permutations $g_{r}:=[\infty,a,b_{1},\infty]$ for $r=1,\dots,\lambda$. Suppose first that $g_{1}$ is trivial. Observe that $(b_{r},c_{r})$ is a transposition in $[\infty,a]$, and, according to supersimplicity $b_{r}\in\ell_{r}$, and only in $\ell_{r}$. So, $b_{r}\in\overline{a,b_{1}}$ or $b_{r}\in\overline{b_{1},\infty}$. Then, $g_{r}:=[\infty,a,b_{r},\infty]$ is trivial, also. Changing $b_{1}$ with $b_{r}$ and $b_{r}$ with $b_{1}$ we obtain that $g_{r}$ is trivial if and only if $g_{1}$ is trivial.

Now, according to Theorem 4.1 we can choose $a\in\Omega\smallsetminus\{\infty\}$ so that $g_{1}$ is not trivial. Thus the same is true for $g_{r}$, for $r=2,3,\ldots,\lambda$. Note, moreover, that the support of $g_{i}$ has at most $6(\lambda-1)$ elements, for $r=1,\dots,\lambda$. Then, since (1) is not true, we conclude that $|\operatorname{supp}(g_{r})|=6(\lambda-1)$ for $r=2,\ldots,\lambda$. It follows that the elements $g_{r}$ have cycle type $2^{3(\lambda-1)}$. Now, note that $g_{1}$ and $g_{2}$ include the transposition $(b_{3},c_{3})$, and the elements $b_{3},c_{3}$ can’t appear in any other transposition of $g_{1}$ and $g_{2}$, so $g_{1}\neq g_{2}$, and this proves $(2)$. ∎

Lemma 5.6**.**

Let $G:=\pi_{\infty}(\mathcal{D})$ a puzzle group, where $\mathcal{D}:=(\Omega,\mathcal{B})$ is a $2-(17,4,3)$ supersimple design. If $G$ is primitive, then $G$ is not isomorphic to any of the groups $2^{4}:D(2*5)$, $(\operatorname{Alt}(4)\times\operatorname{Alt}(4)):2$, $(2^{4}:5).4$, $\mathrm{AGL}_{1}(16):2$, $\mathrm{A\Upgamma L}_{1}(16)$.

Proof.

A calculation with [GAP19] shows that all non-trivial elements in the listed groups have support at least $6(\lambda-1)=12$. Furthermore, these elements have cycle structure $2^{6}$. Also, for every pair of different elements $g,h$ of this cycle type, we have that every transposition in $g$ is different to every transposition in $h$.

Now, applying Lemma 5.5 we get a contradiction.

∎

Combining this result with the earlier restrictions given in Lemma 5.3, we find that there are precisely 14 possible hole-stabilizers when $n=17$. In the GAP library, they are PrimitiveGroup(16,i) for $i\in\{7,8,10,11,12,13,14,15,16,17,18,19,20,21\}$. The case $i=21$ corresponds to $\operatorname{Alt}(16)$.

More generally, for $n\leqslant 17$, the only known example when $G$ is primitive and not equal to $\operatorname{Alt}(n-1)$ occurs when $n=16$, and $G\cong\operatorname{Sym}(6)={\rm Sp}_{4}(2)$. This example was first noted in [GGNS16], and then generalized to an infinite family in [GGS18].

5.4. The case when $\lambda=4$

It would be interesting to see if it might be possible to extend the classification to include puzzle groups arising from supersimple $2-(n,4,4)$ designs. Note, first, that such designs only occur for $n\equiv 1\pmod{3}$ and $n\geqslant 10$.

If $n=10$, an easy counting argument confirms that a supersimple $2-(10,4,4)$ design is also a $3-(10,4,1)$ design. Now a computer calculation by Professor Brendan McKay confirms that there is only one such design, and its hole stabilizer is $\operatorname{Sym}(9)$. For the record, this design has 30 lines as follows:

[TABLE]