Classifying Finite Simple Groups with Respect to the Number of Orbits Under the Action of the Automorphism Group

Abstract

Let ω(G) denote the number of orbits on the finite group G under the action of Aut(G). Using the classification of finite simple groups, we prove that for any positive integer n, there is only a finite number of (non-abelian) finite simple groups G satisfying ω(G) ≤ n. Then we classify all finite simple groups G such that ω(G) ≤ 17. The latter result was obtained by computational means, using the computer algebra system GAP.     

幂零Hall π-子群的存在性

本文首先将Ｈａｌ定理推广为：设Ｎ为Ｇ的正规子群,若Ｎ为Ｅｎπ群,Ｇ／Ｎ为Ｄπ群,则Ｇ为Ｄπ群．在此基础上得到了群Ｇ为Ｅｎπ群的充要条件为：（１）Ｇ存在正规子群Ｎ,满足Ｎ及Ｇ／Ｎ为Ｅｎπ群;（２）对任意ｐ∈π,任意ｑ∈π ｛ｐ｝及任意ｐ 元素ｘ,ＣＧ（ｘ）含Ｇ的Ｓｙｌｏｗｑ 子群．另外,我们对非Ａｂｌｅ单群的情形也进行了一些讨论．     

On sse-embedded subgroups of finite groups

Abstract

In this paper, we introduce the concept of sse-embedded subgroups of finite groups and present some new characterizations of solubility of finite groups using the sse-embedding property of subgroups. Furthermore, we discuss the sse-embedded subgroups in finite nonabelian simple groups. Some previously known results are generalized and unified.     

Affine distance-transitive graphs and classical groups

This paper finishes the classification of the finite primitive affine distance-transitive graphs by dealing with the only case left open, namely where the generalized Fitting subgroup of the stabilizer of a vertex is modulo scalars a simple group of classical Lie type defined over the characteristic dividing the number of vertices of the graph. All graphs that are found to occur are known.     

On Permutation Groups of Degree a Product of Two Prime-Powers

We determine finite simple groups which have a subgroup of index with exactly two distinct prime divisors. Then from this we derive a classification of primitive permutation groups of degree a product of two prime-powers.     

Nonsoluble length of finite groups with restrictions on Sylow subgroups

By 𝔛(n) we denote the variety of all groups satisfying the law [x,y]ⁿ≡1, that is, groups with commutators of order dividing n. Let p be a prime and G a finite group whose Sylow p-subgroups have normal series of length k all of whose quotients belong to 𝔛(n). We show that the non-p-soluble length λ_p(G) of G is bounded in terms of k and n only (Theorem 1.2). In the case where p is odd, a stronger result is obtained (Theorem 1.3).     

Generation and random generation: From simple groups to maximal subgroups

Let G   be a finite group and let d(G)$d(G)$ be the minimal number of generators for G  . It is well known that d(G)=2$d(G)=2$ for all (non-abelian) finite simple groups. We prove that d(H)?4$d(H)?4$ for any maximal subgroup H of a finite simple group, and that this bound is best possible.     

On Thompson's Conjecture of A 10

In this short article, we prove the Thompson's Conjecture is true for the alternating group A ₁₀, where A ₁₀ has exactly one connected prime graph component.     

The locally 2-arc transitive graphs admitting an almost simple group of Suzuki type

In this paper, seven families of vertex-intransitive locally (G,2)-arc transitive graphs are constructed, where Sz(q)?G?Aut(Sz(q)), q=²2k+1 for some k∈N. It is then shown that for any graph Γ in one of these families, Sz(q)?Aut(Γ)?Aut(Sz(q)) and that the only locally 2-arc transitive graphs admitting an almost simple group of Suzuki type whose vertices all have valency at least three are (i) graphs in these seven families, (ii) (vertex transitive) 2-arc transitive graphs admitting an almost simple group of Suzuki type, or (iii) double covers of the graphs in (ii). Since the graphs in (ii) have been classified by Fang and Praeger (1999) [6], this completes the classification of locally 2-arc transitive graphs admitting a Suzuki simple group     

On the probability that finite spaces with random distances are metric spaces

For a set of 3 or 4 points we compute the exact probability that, after assigning the distances between these points uniformly at random from the set 1,…,n , the space obtained is metric. The corresponding results for random real distances follow easily. We also prove estimates for the general case of a finite set of points with uniformly random real distances.     

On Regular Orbits of Elements of Classical Groups in Their Permutation Representations

Let H be a simple finite classical group not isomorphic to PSL(n, q) for any n, q. We prove that every cyclic subgroup of H has a regular orbit on every nontrivial permutation H-set.