Cayley Snarks and Almost Simple Groups

An m-Cayley graph $$\Gamma $$ over a group G is defined as a graph which admits G as a semi-regular group of automorphisms with m orbits. This generalises the notions of a Cayley graph (where $$m = 1$$ ) and a bi-Cayley graph (where $$m = 2$$ ). The m-Cayley graph $$\Gamma $$ over G is said to be normal if G is normal in the automorphism group $$\mathrm{Aut}(\Gamma )$$ of $$\Gamma $$ , and core-free if the largest normal subgroup of $$\mathrm{Aut}(\Gamma )$$ contained in G is the identity subgroup. In this paper, we investigate properties of symmetric m-Cayley graphs in the special case of valency 3, and use these properties to develop a computational method for classifying connected cubic core-free symmetric m-Cayley graphs. We also prove that there is no 3-arc-transitive normal Cayley graph or bi-Cayley graph (with valency 3 or more), which answers a question posed by Li (Proc Amer Math Soc 133:31–41 2005). Using our classification method, we give a new proof of the fact that there are exactly 15 connected cubic core-free symmetric Cayley graphs, two of which are Cayley graphs over non-abelian simple groups. We also show that there are exactly 109 connected cubic core-free symmetric bi-Cayley graphs, 48 of which are bi-Cayley graphs over non-abelian simple groups, and that there are 1, 6, 81, 462 and 3267 connected cubic core-free 1-arc-regular 3-, 4-, 5-, 6- and 7-Cayley graphs, respectively.

2-Arc-transitive Cayley graphs on alternating groups

Throughout this note, G will be a finite group, IrrðGÞ will be the set of irreducible characters of G, cdðGÞ will be the set of character degrees of G, and rðGÞ will be the set of primes that divide degrees in cdðGÞ. The prime vertex degree graph of G, written DðGÞ, is the graph with rðGÞ as its vertex set, and with an edge between p and q if pq divides some degree a A cdðGÞ. An overview of the literature on these graphs can be found in [2]. It seems that most groups G have graphs DðGÞ that are complete graphs (although we do not want to be precise on what we mean by ‘most’). For solvable groups, it has been shown that if DðGÞ is not complete, then the structure of G is limited (see [4] and [6]). In this note, we show that a large class of non-solvable groups have degree graphs that are complete. To study the degree graphs of non-solvable groups, a standard starting point is with the degree graphs of non-abelian simple groups. In a series of papers, Don White has classified the character degree graphs of all non-abelian simple groups. Except for some examples of small rank, nearly all non-abelian simple groups have degree graphs that are complete graphs. These results are summarized in [5]. Extending this idea, this note looks at the characteristically simple groups, groups for which there are exactly two characteristic subgroups. Each such group is a direct product of copies of a fixed simple group. With this in mind, we let G denote G G (n copies); thus S is characteristically simple when S itself is a simple group. We now fix S to be a non-abelian simple group. Because the non-abelian simple groups have already been addressed, we restrict our attention to when n > 1. However, it is relatively easy to show that DðSÞ is complete for n > 1. More generally, the graph DðHÞ associated with any non-abelian group H is always complete for every integer n > 1. Thus we expand our scope and consider extending these groups. We recognize first that S is isomorphic to InnðSÞ. Identifying S with its set of inner automorphisms, our goal herein is to prove the following result.

Symmetric graphs of valency five and their basic normal quotients

In a graph $\Gamma$ with vertex set $V$, a subset $C$ of $V$ is called an $(a,b)$-perfect set if every vertex in $C$ has exactly $a$ neighbors in $C$ and every vertex in $V\setminus C$ has exactly $b$ neighbors in $C$, where $a$ and $b$ are nonnegative integers. In the literature $(0,1)$-perfect sets are known as perfect codes and $(1,1)$-perfect sets are known as total perfect codes. In this paper we prove that, for any finite group $G$, if a non-trivial normal subgroup $H$ of $G$ is a perfect code in some Cayley graph of $G$, then it is also an $(a,b)$-perfect set in some Cayley graph of $G$ for any pair of integers $a$ and $b$ with $0\leqslant a\leqslant|H|-1$ and $0\leqslant b\leqslant |H|$ such that $\gcd(2,|H|-1)$ divides $a$. A similar result involving total perfect codes is also proved in the paper.

A Cayley graph of a group $H$ is a finite simple graph $\Gamma$ such that ${\rm Aut}(\Gamma)$ contains a subgroup isomorphic to $H$ acting regularly on $V(\Gamma)$, while a Haar graph of $H$ is a finite simple bipartite graph $\Sigma$ such that ${\rm Aut}(\Sigma)$ contains a subgroup isomorphic to $H$ acting semiregularly on $V(\Sigma)$ and the $H$-orbits are equal to the bipartite sets of $\Sigma$. A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that the groups $D_6, \, D_8, \, D_{10}$ and $Q_8$ are the only finite inner abelian groups all of whose Haar graphs are Cayley graphs (a group is called inner abelian if it is non-abelian, but all of its proper subgroups are abelian). As an application, it is also shown that every non-solvable group has a Haar graph which is not a Cayley graph.

The workshop Permutation groups organised by Robert Guralnick (Southern California), Cheryl Praeger (Western Australia), Jan Saxl (Cambridge) and Katrin Tent (Bielefeld) was held August 5th–11th, 2007. The focus was recent developments in permutation group theory and their influence on, and from, group representation theory, algebraic graph theory and other areas of geometry and combinatorics, algebraic geometry, model theory and infinite symmetrical structures. It was well attended with 50 participants. Especially valuable was the program to support young researchers which gave the opportunity for a considerable number of young mathematicians to participate in the workshop. They benefited from the ‘Oberwolfach experience’ and also enlivened and contributed enormously to the success of the meeting. The workshop had a friendly and vibrant atmosphere, aided by the excellent facilities of the Institute. All participants appreciated the beautiful lecture by John G. Thompson entitled ‘The divisor matrix and {\rm SL}(2,\Z) ’. (In fact, this meeting was the first Oberwolfach workshop in over twenty years that Thompson had attended.) There were 26 other talks, and the program featured in each of the first five sessions at least one talk by a student or young researcher at the beginning of their career. This facilitated interaction between younger and more established researchers. The number of talks was restricted to give plenty of time for discussion and cooperation among the participants. Other communications were presented as posters. Highlights of the workshop addressed both fundamental permutation group theory, and also significant outcomes from applying permutation group theory and methods in combinatorics, model theory and other areas. Some topics arose in more than one context, forming new connections. A common feature of many contributions was application of the finite simple group classification, or new results on the structure of simple groups. As examples, we mention two highlights reported by early career participants. The question of base size of permutation actions is of importance in computational group theory as well as in the study of the graph isomorphism problem. Recent research has thrown much light on the base sizes of actions of almost simple groups in particular. Burness reported on the very recent proof of a conjecture of Cameron and Kantor dating from 1993 concerning non-standard permutation actions of almost simple groups: these are all primitive actions apart from actions of alternating and symmetric groups on subsets or partitions, and of classical groups on subspaces or pairs of complementary subspaces. In all the latter families the minimum base size is unbounded as the group order increases. The somewhat surprising conjecture, now proved, is that the minimum base size of non-standard permutation groups is at most 7, with the unique example where this bound is attained being the Mathieu group M_{24} in its natural action on 24 points. Expander graphs play an important role in computer science and combinatorics, and, for example, are significant in modelling and analysing communication networks. In particular, their diameters grow only logarithmically with the numbers of vertices. This property on diameters was proved in 1989 by Babai, Kantor and Lubotzky to hold for suitable Cayley graphs of simple groups: specifically for each finite nonabelian simple group G , they constructed a valency 7 Cayley graph for G with diameter at most a constant times \log(|G|) . Although their graphs were not expanders, they conjectured that there exists an absolute constant k such that all infinite families of simple groups give rise to families of valency k Cayley graphs that are expanders. Kassabov lectured on amazing progress towards proving this conjecture – it is now proved for all simple groups apart from the Suzuki groups. It was a very happy and successful workshop. Indeed, several of the most distinguished participants commented that this was one of the best Oberwolfach meetings they had attended, with plenty of time for collaboration, and outstanding talks. Extended abstracts of the talks mentioned, as well as all the others, are given below (in the order in which they were presented).

Let G be a nonabelian group and associate a noncommuting graph ∇(G) with G as follows: The vertex set of ∇(G) is G\\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In 1987, Professor J. G. Thompson gave the following conjecture. Thompson's Conjecture. If G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying N(G) = N(M), then G ≅ M, where N(G):={n ∈ ℕ | G has a conjugacy class of size n}. In 2006, A. Abdollahi, S. Akbari, and H. R. Maimani put forward a conjecture (AAM's conjecture) in Abdollahi et al. (2006) as follows. AAM's Conjecture. Let M be a finite nonabelian simple group and G a group such that ∇(G) ≅ ∇ (M). Then G ≅ M. In this short article we prove that if G is a finite group with ∇(G) ≅ ∇ (A 10), then G ≅ A 10, where A 10 is the alternating group of degree 10.

Integral Cayley graphs over dicyclic group

1·1. We shall write for the class of groups all of whose non-trivial normal subgroups have finite index. Thus, rather obviously, finite groups, simple groups, and the infinite cyclic and dihedral groups all lie in the class . Other examples of -groups are to be found in a variety of contexts. The main result of Mennicke (8) shows that, for n ≥ 3, the factor group of SL(n, Z) by its centre is a -group, where SL(n, Z) denotes the unimodular group of all n × n invertible matrices with integer coefficients and with determinant 1. In McLain(7), an example is given of an infinite, locally finite, locally soluble -group, whose only non-trivial normal subgroups are the terms of its derived series, and in (4), P. Hall discusses an infinite -group, all of whose proper quotients are finite p-groups. If is a class of groups closed under taking homomorphic images, it is easily seen that the existence of an infinite -group satisfying the maximal condition for normal subgroups implies the existence of an infinite -group in the class , so that certain questions concerning the finiteness of groups satisfying the maximal condition for normal subgroups can be interpreted as questions about -groups.

Let G be a finite nonabelian group and associate a disoriented noncommuting graph ∇(G) with G as follows: the vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In 1987, J. G. Thompson gave the following conjecture.Thompson's Conjecture If G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying N(G) = N(M), then G ≅ M, where N(G) denotes the set of the sizes of the conjugacy classes of G.In 2006, A. Abdollahi, S. Akbari and H. R. Maimani put forward a conjecture in [1] as follows.AAM's Conjecture Let M be a finite nonabelian simple group and G a group such that ∇(G)≅ ∇ (M). Then G ≅ M.Even though both of the two conjectures are known to be true for all finite simple groups with nonconnected prime graphs, it is still unknown for almost all simple groups with connected prime graphs. In the present paper, we prove that the second conjecture is true for the projective special unitary simple group U4(7).

All groups considered are finite. A group has a trivial Frattini subgroup if and only if every nontrivial normal subgroup has a proper supplement.The property is normal subgroup closed, but neither subgroup nor quotient closed. It is subgroup closed if and only if the group is elementary, i.e. all Sylow subgroups are elementary abelian. If G is solvable, then G and all its quotients have trivial Frattini subgroup if and only if every normal subgroup of G has a complement. For a nilpotent group, every nontrivial normal subgroup has a supplement if and only if the group is elementary abelian. Consequently, the center of a group in which every normal subgroup has a supplement is an elementary abelian direct factor.

For a finite group G, let the character degree set of G, denoted by $$\mathrm{cd}(G)$$, be the set of the degrees of all irreducible complex representations of G. In the present paper, the structure of finite groups whose character degree sets coincide with the character degree sets of direct product of nonabelian simple groups will be studied. For a finite group S and a positive integer n, let $$S^{n}$$ be the direct product of n copies of S. We prove that if G is a finite group with $$\mathrm{cd}(G)=\mathrm{cd}(H)$$, where $$\begin{aligned}&H\in \{\mathrm{PSL}_{3}(q)^{n}(q=2^{\alpha }\geqslant 4, 3\not \mid q-1), \mathrm{PSU}_{3}(q)^{n}(q=2^{\alpha }\geqslant 4, 3\not \mid q+1), \\&\quad M_{11}^{n} (1\leqslant n\leqslant 2), M_{23}^{n}, J_{1}^{n}, J_{2}^{n}, J_{3}^{n}, J_{4}^{n}\}, \end{aligned}$$then G is a quasi perfect group. This extends the first step of Huppert’s Conjecture to the direct product of simple groups. This conjecture states that the nonabelian simple groups are uniquely determined up to an abelian direct factor by the set of character degrees.

In 2004, Fang, Lu, Wang and Xu proved that, with few exceptions, connected Cayley digraphs of finite nonabelian simple groups with out-valency two are normal, enabling a complete determination of the automorphism groups of such Cayley digraphs. In this paper, we present significant improvements: we not only reduce the list of exceptions but also extend these results to the more general setting of products of nonabelian simple groups. This includes the case of nonabelian characteristically simple groups, which is crucial in Praeger’s approach to analyzing the automorphism group of a general Cayley digraph.

In answer to a question of P. Hall, we supply another construction of a group which is isomorphic to each of its non-trivial normal subgroups.