Cubic graphical regular representations of Ree groups

Abstract

A graphical regular representation of a group G is a Cayley graph of G whose full automorphism group is equal to the right regular permutation representation of G. In this paper, we prove that for Ree groups $\text{Ree}(q)$ with q > 3, with probability tending to 1 as $q\to \infty$, a random involution y together with a fixed element x with order q – 1 gives rise to a cubic graphical regular representation of $\text{Ree}(q)$. A similar result involving a fixed element with order $q+\sqrt{3q}+1$ is also proved with the help of certain properties of $\text{Ree}(q)$ given in [Leemans, D. Liebeck, M. W. (2017). Chiral polyhedra and finite simple groups. Bull. London Math. Soc. 49: 581–592].

KEYWORDS:

• Cayley graph
• graphical regular representation
• Ree group

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 05C25
• 05C30
• 20B25
• 20D06

1 Introduction

All groups considered in this paper are finite, and all graphs considered are finite and undirected. Let G be a group whose identity element is denoted as 1, and let S be a subset of G such that $1\notin S$ and $S^{-1}=S$, where $S^{-1}=\{x^{-1}:x\in S\}$. The Cayley graph of G with connection set S, denoted by $\text{Cay}(G,S)$, is defined as the graph with vertex set G such that x and y are adjacent if and only if $y{x}^{-1}\in S$. It is well known that the right regular permutation representation of G is a subgroup of the automorphism group of $\text{Cay}(G,S)$. It is also widely known that a graph Γ is isomorphic to a Cayley graph if and only if its automorphism group $\text{Aut}(\Gamma )$ contains a subgroup which is regular on the vertex set of Γ. A graph Γ is said [4] to be a graphical regular representation (GRR) of a group G if $\text{Aut}(\Gamma )$ is isomorphic to G and is regular on the vertex set of Γ. Thus, up to isomorphism, a GRR of a group G is precisely a Cayley graph of G whose automorphism group is equal to the right regular permutation representation of G. If there exists some Cayley graph which is a GRR of G, then G is said to admit a GRR.

It is natural to ask which finite groups admit GRRs. This question was studied in a series of papers (see, for example, [2, 12, 15, 17]), and eventually a complete characterization was obtained by Godsil in [4]: Apart from abelian groups of exponent at least three, generalized dicyclic groups and thirteen other groups, every finite group admits a GRR. There is also special interest in studying which finite groups admit GRRs of a prescribed valency (see, for example, [3, 5, 16, 19–21, 23]). In the case of valency three, Fang, Li, Wang and Xu [3] conjectured that every finite non-abelian simple group admits a cubic GRR. However, in [19], Xia and Fang found that this conjecture fails for ${\text{PSL}}_2(7)$. Meanwhile, they proposed the following conjecture in the same paper.

Conjecture 1.1.

[19, Conjecture 4.3] Except for a finite number of cases, every finite non-abelian simple group has a cubic GRR.

It is known that if $\text{Cay}(G,S)$ is a GRR of G with more than two vertices, then we necessarily have $\text{Aut}(G,S)=1$ and $G=〈S〉$, where $\text{Aut}(G,S)=\{\alpha \in \text{Aut}(G):S^{\alpha }=S\}$. With the help of [3, Theorem 1.3] and [16, Theorem 1.1], a list of finite non-abelian simple groups was given in [20] such that, for any group G in this list, a cubic Cayley graph $\text{Cay}(G,S)$ is a GRR if and only if $\text{Aut}(G,S)=1$ and $G=〈S〉$. Among other groups this list contains all sporadic simple groups, all simple groups of exceptional Lie type, some alternating groups, and some classical groups. On the other hand, in [7, Theorem 1.1], Leemans and Liebeck proved that for every finite non-abelian simple group G except A₇, ${\text{PSL}}_2(q),{\text{PSL}}_3(q)$ and ${\text{PSU}}_3(q)$, there exists a pair of generators (x, y) of G where y is an involution such that $\text{Aut}(G,\{x,x^{-1},y\})=1$. Combining this with the above-mentioned list in [20], we see that most families of finite non-abelian simple groups can be proved to have cubic GRRs. In particular, all sporadic simple groups and simple groups of exceptional Lie type admit cubic GRRs. Alternating groups of degree at least 19 [5] and some classical groups [19–21] are also known to admit cubic GRRs. Thus, to settle Conjecture 1.1, the remaining families of groups that need to be considered are: ${\text{PSU}}_3(q)$ with $q\ne 2,{\text{PSL}}_4(q),{\text{PSL}}_6(q)$ where $\gcd (6,q-1)=1,{\text{PSL}}_8(q),{\text{PSp}}_6(q)$, ${\text{PSp}}_8(q),\mathrm{P}{\Omega }_8^{\pm }(q),\mathrm{P}{\Omega }_{10}^{\pm }(q)$, and $\mathrm{P}{\Omega }_{12}^{\pm }(q)$ with q even. During the preparation of this paper, cubic GRRs of these classical groups above are studied in [9, 22], leading to the confirmation of Conjecture 1.1 in [22].

Besides the existence of cubic GRRs, there is also considerable interest in the proportion of cubic Cayley graphs of a given group that are GRRs. In [16, Conjecture 1.4], Spiga conjectured that, for finite non-abelian simple groups, this proportion approaches 1 as the order of the group approaches infinity. Evidence in favor of this conjecture for some simple classical groups can be found in [19, Theorem 1.4] and [21, Theorem 1.3]. In this paper we present two results for Ree groups in support of Spiga’s conjecture. It is well known that the Ree groups $\text{Ree}(q)$ [14] with q > 3 are simple groups of exceptional Lie type. So they admit cubic GRRs as per discussion in the previous paragraph. Inspired by [21, Theorem 1.3], we prove further that almost every involution of $\text{Ree}(q)$ together with a fixed element of order q – 1 gives rise to a cubic GRR of $\text{Ree}(q)$. In what follows a random involution is meant to be an involution chosen uniformly at random from all involutions of $\text{Ree}(q)$.

Theorem 1.2.

Let $G=\text{Ree}(q)$, where $q=3^{2n+1}$ with $n\ge 1$, and let x be any fixed element of G with order q – 1. Then for a random involution y of G the probability of $\text{Cay}(G,\{x,x^{-1},y\})$ being a GRR of G is at least $1-q^{-1}-2{(q^2-q+1)}^{-1}$.

Upon proving this result, we realized that a few properties of Ree groups established in the proof of [7, Theorem 1.1] imply the following result.

Theorem 1.3.

Let $G=\text{Ree}(q)$, where $q=3^{2n+1}$ with $n\ge 1$, and let x be any fixed element of G with order $q+\sqrt{3q}+1$. Then for a random involution y of G the probability of $\text{Cay}(G,\{x,x^{-1},y\})$ being a GRR of G is at least $1-q^{-1}-\sqrt{3}{q}^{-3/2}-q^{-2}$.

Since both $1-q^{-1}-2{(q^2-q+1)}^{-1}$ and $1-q^{-1}-\sqrt{3}{q}^{-3/2}-q^{-2}$ approach 1 as q approaches infinity, Theorems 1.2 and 1.3 provide evidence in some sense (without considering connection sets of size three other than those considered in these two theorems) to support [16, Conjecture 1.4] for Ree groups.

Theorem 1.2 will be proved in Section 3 after recollecting some known results on Ree groups in the next section. Theorem 1.3 will be proved in Section 4 in a similar way as the proof of Theorem 1.2.

2 Preliminaries

The notation used in the paper is standard and can be found in [18]. We use ${\mathrm{E}}_q$ to denote an elementary abelian group of order q and $D_{2n}$ a dihedral group of order 2n. For a set A of a group and a positive integer m, denote by $I_m(A)$ the set of elements of order m in A, and set $i_m(A)=|I_m(A)|$.

All maximal subgroups of Ree groups were determined independently in [8] and [6]. We present them in the following theorem (see [18, Theorem 4.2]).

Theorem 2.1.

Let $q=3^{2n+1}$, where $n\ge 1$. The maximal subgroups of $\text{Ree}(q)$ are exhausted, up to conjugacy, by the following:

1. $E_q.E_q.E_q:C_{q-1}$, the normalizer of a Sylow 3-subgroup;

2. $C_2\times PS{L}_2(q)$, the centralizer of an involution;

3. $(C_2\times C_2\times D_{(q+1)/2}):C_3$, the normalizer of a cyclic Hall subgroup of order $(q+1)/4$;

4. $C_{q+\sqrt{3q}+1}:C_6$, the normalizer of a cyclic Hall subgroup of order $q+\sqrt{3q}+1$;

5. $C_{q-\sqrt{3q}+1}:C_6$, the normalizer of a cyclic Hall subgroup of order $q-\sqrt{3q}+1$;

6. $\text{Ree}(q_0)$, where $q=q_0{}^r$ and r is prime.

The following results are collected from [18, Section 4.5.3], [11, Theorem 3.12], [1, Proposition 3.3], [13, Lemma 3.13] and [16, Theorem 1.1]. Note that part (ii) follows from [11, Theorem 3.12] because all involutions in $\text{Ree}(q)$ are conjugate.

Proposition 2.2.

Let G be the Ree group $\text{Ree}(q)$, where $q=3^{2n+1}$ with $n\ge 1$. The following hold for G and its maximal subgroups.

1. The outer automorphism group of G is $C_{2n+1}$.

2. $i_2(G)=q^2(q^2-q+1)$; $i_2(M_1)=q^2$ for any maximal subgroup M₁ of the form $E_q.E_q.E_q:C_{q-1}$; $i_2(M_2)=q^2-q+1$ for any maximal subgroup M₂ of the form $C_2\times {\text{PSL}}_2(q)$.

3. The intersection of any two distinct maximal subgroups of the form $E_q.E_q.E_q:C_{q-1}$ is $C_{q-1}$; the intersection of any three distinct maximal subgroups of G of the form $E_q.E_q.E_q:C_{q-1}$ has order at most two.

4. Any two distinct maximal subgroups of the form $C_2\times {\text{PSL}}_2(q)$ have no common element of order q – 1.

5. Let S be a 3-subset of G such that $1\notin S$ and $S^{-1}=S$. If $〈S〉=G$ and $\text{Aut}(G,S)=1$, then $\text{Cay}(G,S)$ is a GRR of G.

3 Proof of Theorem 1.2

Let G be the Ree group $\text{Ree}(q)$, where $q=3^{2n+1}$ with $n\ge 1$. Fix $x\in I_{q-1}(G)$. Define $$K=\{y\in I_2(G):G=〈x,y〉\}, L=\{y\in I_2(G):\text{Aut}(G,\{x,x^{-1},y\})=1\},$$and let $\mathcal{M}$ be the set of maximal subgroups of G containing x.

As mentioned earlier, if $\text{Cay}(G,S)$ is a GRR of G, then $\text{Aut}(G,S)=1$ and $G=〈S〉$. This combined with Proposition 2.2(v) implies that for $y\in I_2(G)$, the Cayley graph $\text{Cay}(G,\{x,x^{-1},y\})$ is a GRR of G if and only if $y\in K\cap L$. Thus for a random involution y of G, the probability of $\text{Cay}(G,\{x,x^{-1},y\})$ being a GRR of G is given by (3.1) $$P=\frac{|K\cap L|}{i_2(G)}.$$(3.1)

It is clear that $$I_2(G)\setminus K=\{y\in I_2(G):G\ne 〈x,y〉\}=I_2\left(\underset{M\in \mathcal{M}}{\cup }M\right).$$

Thus (3.2) $$\left|K\right|=i_2(G)-i_2\left(\underset{M\in \mathcal{M}}{\cup }M\right).$$(3.2)

By Theorem 2.1 and Proposition 2.2(iii)(iv), there are at most two different maximal subgroups of the form $E_q.E_q.E_q:C_{q-1}$ and at most one maximal subgroup of the form $C_2\times {\text{PSL}}_2(q)$ containing x. Hence, by Proposition 2.2(ii), we have $$i_2\left(\underset{M\in \mathcal{M}}{\cup }M\right)\le 2q^2+(q^2-q+1).$$

This together with (3.2) and $i_2(G)=q^2(q^2-q+1)$ (Proposition 2.2(ii)) implies that (3.3) $$\left|K\right|\ge (q^2-1)(q^2-q+1)-2q^2.$$(3.3)

Define $$J=\{\alpha \in I_2(\text{Aut}(G)):x^{\alpha }=x^{-1}\}.$$

Since $\text{Aut}(G)=\text{Inn}(G).\text{Out}(G)=\text{Inn}(G).C_{2n+1}$, we have $J\subseteq \text{Inn}(G)$.

By the definitions of K and L, for each $y\in K\setminus L$, there exists a nontrivial automorphism $\alpha \in \text{Aut}(G)$ which fixes $\{x,x^{-1},y\}$ setwise. Note that both x and $x^{-1}$ have order $q-1>2$ while y is an involution. Thus $x^{\alpha }=x^{-1}$ and $y^{\alpha }=y$. Since ${\alpha }^2$ fixes both x and y, and $〈x,y〉=G$, we have $\left|\alpha \right|=2$. This combined with $x^{\alpha }=x^{-1}$ implies $\alpha \in J$. Moreover, by $y^{\alpha }=y$, it is clear that $y\in {\mathbf{C}}_G(\alpha )$. Notice that ${\mathbf{C}}_G(\alpha )=C_2\times {\text{PSL}}_2(q)$ as $\alpha \in J\subseteq \text{Inn}(G)$. Therefore, $$|K\setminus L|\le \sum _{\alpha \in J}{i}_2({\mathbf{C}}_G(\alpha ))\le \left|J\right|(q^2-q+1).$$

Since G is simple, the mapping $\text{Inn}:G\to \text{Inn}(G)$ is one-to-one, where $\text{Inn}(g)$ is the inner automorphism of G given by g. Suppose α₀ is an involutory automorphism in J. For $\alpha \in J$, there exists $g(\alpha )\in {\mathbf{C}}_G(x)$ such that $\alpha =\text{Inn}(g(\alpha )){\alpha }_0$. It follows that $\left|J\right|\le |{\mathbf{C}}_G(x)|$. By [10, Lemma 2.6], we know ${\mathbf{C}}_G(x)=〈x〉$. Hence $\left|J\right|\le q-1$. Thus (3.4) $$|K\setminus L|\le (q-1)(q^2-q+1).$$(3.4)

Finally, by (3.1), (3.3), (3.4) and Proposition 2.2(ii), we obtain $$P=\frac{\left|K\right|-|K\setminus L|}{i_2(G)}\ge 1-\frac{1}{q}-\frac{2}{q^2-q+1}.$$

This completes the proof of Theorem 1.2.

4 Proof of Theorem 1.3

Let $G=\text{Ree}(q)$ be as in the previous section. In [7], Leemans and Liebeck considered a cyclic maximal torus $〈x〉$ of G with order $q+\sqrt{3q}+1$. To prove that G acts as the automorphism group of a chiral polyhedron, they proved the existence of an involution y of G such that y lies in no maximal subgroup of G containing x, and y centralizes no involutory automorphism of G that inverts x. We see from the proof of Theorem 1.2 that the set of such involutions, denoted by $\overline{S}$, is similar to $K\cap L$ in the previous section and that the existence of such involutions implies the existence of cubic GRRs of G. In other words, for a random involution y of G, the probability of $\text{Cay}(G,\{x,x^{-1},y\})$ being a GRR of G is given by $$P=\frac{\left|\overline{S}\right|}{i_2(G)}.$$

The following results were proved in [7, Section 3]:

• the number of involutory automorphisms of G that invert x is no more than the order of the torus;

• the centralizer of such an involutory automorphism in G is still $C_2\times {\text{PSL}}_2(q)$;

• there is no maximal subgroup of G containing x other than $N_G(〈x〉)$.

Thus, $$P\ge 1-\frac{|〈x〉|i_2(C_2\times {\text{PSL}}_2(q))}{i_2(G)}.$$

Since $i_2(G)=q^2(q^2-q+1)$ and $i_2(C_2\times {\text{PSL}}_2(q))=q^2-q+1$ by Proposition 2.2(ii) and $|〈x〉|=q+\sqrt{3q}+1$, we have $$P\ge 1-\frac{q+\sqrt{3q}+1}{q^2}$$as required to complete the proof of Theorem 1.3.

Acknowledgment

The third author was supported by the Melbourne Research Scholarship.