Finite Nonabelian Simple Group Research Articles

On Automorphism Groups of Symmetric Cayley Graphs of Finite Simple Groups with Valency Six

In this paper we investigate the full automorphism groups of six-valent symmetric Cayley graphs Γ = Cay (G,S) for finite non-abelian simple groups G. We prove that for most finite non-abelian simple groups G, if Γ contains no cycle of length 4, then Aut Γ = G · Aut (G,S), where Aut (G,S) ≤ S 6.

Root systems and Weyl groupoids for Nichols algebras

Motivated by work of Kac and Lusztig, we define a root system and a Weyl groupoid for a large class of semisimple Yetter-Drinfeld modules over an arbitrary Hopf algebra. The ob- tained combinatorial structure fits perfectly into an existing frame- work of generalized root systems associated to a family of Cartan matrices, and provides novel insight into Nichols algebras. We demonstrate the power of our construction with new results on Nichols algebras over finite non-abelian simple groups and sym- metric groups.

Zeros of Brauer characters and the defect zero graph

We show that for each finite non-abelian simple group and each prime p either there exists an irreducible Brauer character which takes the value zero on some p-regular element, or p ¼ 2 and all degrees of irreducible Brauer characters are powers of 2. This generalizes an old result of Burnside from ordinary to Brauer characters. For the proof we introduce the notion of the defect zero graph of a finite group, which may be of independent interest.

Verifying Huppert’s Conjecture for 2 G 2(q 2)

Let G be a finite group and cd(G) be the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G ≅ H×A, where A is an abelian group. In this paper, we verify the conjecture for the twisted Ree groups 2G2(q2) for q2 = 32m + 1, m ≥ 1. The argument involves verifying five steps outlined by Huppert in his arguments establishing his conjecture for many of the nonabelian simple groups.

A classification of nonabelian simple 3-BCI-groups

For a finite group G and a subset S ⊆ G (possibly, S contains the identity of G ), the bi-Cayley graph BCay ( G , S ) of G with respect to S is the graph with vertex set G × { 0 , 1 } and with edge set { ( x , 0 ) , ( s x , 1 ) | x ∈ G , s ∈ S } . A bi-Cayley graph BCay ( G , S ) is called a BCI-graph if, for any bi-Cayley graph BCay ( G , T ) , whenever BCay ( G , S ) ≅ BCay ( G , T ) we have T = g S α , for some g ∈ G , α ∈ Aut ( G ) . A group G is called an m-BCI-group, if all bi-Cayley graphs of G of valency at most m are BCI-graphs. In this paper, we prove that a finite nonabelian simple group is a 3-BCI-group if and only if it is A 5 .

On finite simple and nonsolvable groups acting on closed 4–manifolds

We show that the only finite nonabelian simple groups to admit a locally linear, homologically trivial action on a closed simply connected 4-manifold M (or on a 4-manifold with trivial first homology) are the alternating groups A 5 , A 6 and the linear fractional group PSL(2, 7). (We note that for homologically nontrivial actions all finite groups occur.) The situation depends strongly on the second Betti number b 2 (M) of M and was known before if b 2 (M) is different from two, so the main new result concerns the case b 2 (M) = 2. We prove that the only simple group that occurs in this case is A 5 , and then deduce a short list of finite nonsolvable groups which contains all candidates for actions of such groups.

Verifying Huppert's Conjecture for PSL3(q) and PSU3(q 2)

Let G be a finite group and cd(G) be the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G ≅ H × A, where A is an abelian group. We examine arguments to verify this conjecture for the simple groups of Lie type of rank two. To illustrate our arguments, we extend Huppert's results and verify the conjecture for the simple linear and unitary groups of rank two.

On the clique number of the generating graph of a finite group

The generating graph $\Gamma (G)$ of a finite group $G$ is the graph defined on the elements of $G$ with an edge connecting two distinct vertices if and only if they generate $G$. The maximum size of a complete subgraph in $\Gamma (G)$ is denoted by $\omega (G)$. We prove that if $G$ is a non-cyclic finite group of Fitting height at most $2$ that can be generated by $2$ elements, then $\omega (G) = q+1$, where $q$ is the size of a smallest chief factor of $G$ which has more than one complement. We also show that if $S$ is a non-abelian finite simple group and $G$ is the largest direct power of $S$ that can be generated by $2$ elements, then $\omega (G) \leq (1+o(1))m(S)$, where $m(S)$ denotes the minimal index of a proper subgroup in $S$.

Cubic [formula omitted]-arc transitive Cayley graphs

This paper gives a characterization of connected cubic s -transitive Cayley graphs. It is shown that, for s ≥ 3 , every connected cubic s -transitive Cayley graph is a normal cover of one of 13 graphs: three 3-transitive graphs, four 4-transitive graphs and six 5-transitive graphs. Moreover, the argument in this paper also gives another proof for a well-known result which says that all connected cubic arc-transitive Cayley graphs of finite non-abelian simple groups are normal except two 5-transitive Cayley graphs of the alternating group A 47 .

On recognition of finite simple groups with connected prime graph

The spectrum of a finite group is the set of its element orders. We prove a theorem on the structure of a finite group whose spectrum is equal to the spectrum of a finite nonabelian simple group. The theorem can be applied to solving the problem of recognizability of finite simple groups by spectrum.

Recognition of some simple groups by their noncommuting graphs

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. Abdollahi et al. (J Algebra 298(2):468–492, 2006) put forward a conjecture called AAM’s Conjecture in as follows: If M is a finite nonabelian simple group and G is a group such that ∇(G) ≅ ∇(M), then G ≅ M. Even though this conjecture is well known to hold for all simple groups with nonconnected prime graphs and the alternating group A10 [see Darafsheh (Groups with the same non-commuting graph. Discrete Appl Math (2008) doi:10.1016/j.dam.2008.06.010), Wang and Shi (Commun Algebra 36(2):523–528, 2008)], it is still unknown for all simple groups with connected prime graphs except A10. In the present paper, we prove that this conjecture is also true for the projective special linear simple group L4(9). The new method used in this paper also works well in the cases L4(4), L4(7), U4(7), etc.

A NEW CHARACTERIZATION OF U4(7) BY ITS NONCOMMUTING GRAPH

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).

Connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups

A graph is said to be s-arc-regular if its full automorphism group acts regularly on the set of its s-arcs. In this paper, we investigate connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups. Two sufficient and necessary conditions for such graphs to be 1- or 2-arcregular are given and based on the conditions, several infinite families of 1- or 2-arc-regular cubic Cayley graphs of alternating groups are constructed.

A COMPARISON OF THE ORDER COMPONENTS IN FROBENIUS AND 2-FROBENIUS GROUPS WITH FINITE SIMPLE GROUPS

Let $G$ be a finite group. Based on the Gruenberg-Kegel graph ${\rm GK}(G)$, the order of $G$ can be divided into a product of coprime positive integers. These integers are called the order components of $G$ and the set of order components is denoted by ${\rm OC}(G)$. In this article we prove that, if $S$ is a non-Abelian finite simple group with a disconnected graph ${\rm GK}(S)$, with an exception of $U_4(2)$ and $U_5(2)$, and $G$ is a finite group with ${\rm OC}(G)={\rm OC}(S)$, then $G$ is neither Frobenius nor $2$-Frobenius. For a group $S$ isomorphic to $U_4(2)$ or $U_5(2)$, we construct examples of $2$-Frobenius groups $G$ such that ${\rm OC}(S)={\rm OC}(G)$. In particular, the simple groups $U_4(2)$ and $U_5(2)$ are not recognizable by their order components.

An interesting consequence of the Heisenberg construction

The projective linear groups over finite fields give families of nonabelian finite simple groups and it is natural to wonder as to which groups are of the form G/Z(G) for some group G, where Z(G) denotes the center of G. In fact, ideally one would like to characterize all of them. It is natural to believe such a characterization could probably be obtained from the theory of central extensions. Indeed, these groups have been studied under the name of capable groups for some time now. Let us first think about the (far) simpler problem of finite abelian groups which are of this form. The answer turns out to be simple but interesting. Before stating the result, let us notice that the question is about 2-step nilpotent groups, since G/Z(G) is nontrivial, abelian if and only if G is 2-step nilpotent.

Groups with the same non-commuting graph

The non-commuting graph Γ G of a non-abelian group G is defined as follows. The vertex set of Γ G is G − Z ( G ) where Z ( G ) denotes the center of G and two vertices x and y are adjacent if and only if x y ≠ y x . It has been conjectured that if G and H are two non-abelian finite groups such that Γ G ≅ Γ H , then | G | = | H | and moreover in the case that H is a simple group this implies G ≅ H . In this paper, our aim is to prove the first part of the conjecture for all the finite non-abelian simple groups H . Then for certain simple groups H , we show that the graph isomorphism Γ G ≅ Γ H implies G ≅ H .

Combinatorial method in the coset enumeration of symmetrically generated groups

Several finite groups, including all non-abelian finite simple groups, can be generated by finite conjugacy classes of involutions. We describe a coset enumeration algorithm that is particularly developed for a group defined in this manner. We illustrate the process by using some rather small examples for which the enumerations can be worked manually.

Permutation Groups

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).

Finite groups with minimal 1-PIM

Let \(\mathbb F\) be a field of characteristic \(\ell > 0\) and let G be a finite group. It is well-known that the dimension of the minimal projective cover \(\Phi_1^G\) (the so-called 1-PIM) of the trivial left \(\mathbb F[G]\) -module is a multiple of the \(\ell\) -part \(|G|_\ell\) of the order of G. In this note we study finite groups G satisfying \(\dim_{\mathbb F}(\Phi_1^G)=|G|_\ell\) . In particular, we classify the non-abelian finite simple groups G and primes \(\ell\) satisfying this identity (Theorem A). As a consequence we show that finite soluble groups are precisely those finite groups which satisfy this identity for all prime numbers \(\ell\) (Corollary B). Another consequence is the fact that the validity of this identity for a finite group G and for a small prime number \(\ell\in\{2,3,5\}\) implies the existence of an \(\ell^\prime\) -Hall subgroup for G (Theorem C). An important tool in our proofs is the super-multiplicativity of the dimension of the 1-PIM over short exact sequences (Proposition 2.2).

Probabilistic generation of finite simple groups, II

In earlier work it was shown that each nonabelian finite simple group G has a conjugacy class C such that, whenever 1 ≠ x ∈ G , the probability is greater than 1/10 that G = 〈 x , y 〉 for a random y ∈ C . Much stronger asymptotic results were also proved. Here we show that, allowing equality, the bound 1/10 can be replaced by 13/42; and, excluding an explicitly listed set of simple groups, the bound 2/3 holds. We use these results to show that any nonabelian finite simple group G has a conjugacy class C such that, if x 1 , x 2 are nontrivial elements of G, then there exists y ∈ C such that G = 〈 x 1 , y 〉 = 〈 x 2 , y 〉 . Similarly, aside from one infinite family and a small, explicit finite set of simple groups, G has a conjugacy class C such that, if x 1 , x 2 , x 3 are nontrivial elements of G, then there exists y ∈ C such that G = 〈 x 1 , y 〉 = 〈 x 2 , y 〉 = 〈 x 3 , y 〉 . We also prove analogous but weaker results for almost simple groups.