Recognizing Abelian Sylow Subgroups in Finite Groups

Let p be a prime. We prove that if a finite group G has non-abelian Sylow p-subgroups, and the class size of every p-element in G is coprime to p, then G contains a simple group as a subquotient which exhibits the same property. In addition, we provide a list of all the simple groups and primes such that the Sylow p-subgroups are non-abelian and all p-elements have class size coprime to p.     

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.     

Commutativity Pattern of Finite Non-Abelian p-Groups Determine Their Orders

Let G be a non-abelian group and Z(G) be the center of G. Associate a graph Γ _G (called noncommuting graph of G) with G as follows: Take G?Z(G) as the vertices of Γ _G , and join two distinct vertices x and y, whenever xy ≠ yx. Here, we prove that “the commutativity pattern of a finite non-abelian p-group determine its order among the class of groups"; this means that if P is a finite non-abelian p-group such that Γ _P  ? Γ _H for some group H, then |P| = |H|.     

How often can a finite group be realized as a Galois group over a field?

Let G be any finite group and any class of fields. By we denote the minimal number of realizations of G as a Galois group over some field from the class . For G abelian and the class of algebraic extensions of ℚ we give an explicit formula for . Similarly we treat the case of an abelian p-group G and the class which is conjectured to be the class of all fields of characteristic ≠p for which the Galois group of the maximal p-extension is finitely generated. For non-abelian groups G we offer a variety of sporadic results. Received: 27 October 1998 / Revised version: 3 February 1999     

A decomposition theorem for G-groups

Some classical results about linear representations of a finite group G have been also proved for representations of G on non-abelian groups （G-groups）. In this paper we establish a decomposition theorem for irreducible G-groups which expresses a suitable irreducible G-group as a tensor product of two projective G-groups in a similar way to the celebrated theorem of Clifford for linear representations. Moreover, we study the non-abelian minimal normal subgroups of G in which this decomposition is possible.     

On random permutations of finite groups

Given a finite abelian group G,  consider a uniformly random permutation of the set of all elements of G. Compute the difference of each pair of consecutive elements along the permutation. What is the number of occurrences of \(h\in G\setminus \{0\}\) in this sequence of differences? How do these numbers of occurrences behave for several group elements simultaneously? Can we get similar results for non-abelian G? How do the answers change if differences are replaced by sums? In this paper, we answer these questions. Moreover, we formulate analogous results in a general combinatorial setting.

     

On the diameter of the intersection graph of a finite simple group

Let G be a finite group. The intersection graph Δ_G of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G, and two distinct vertices X and Y are adjacent if X ∩ Y ≠ 1, where 1 denotes the trivial subgroup of order 1. A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters of intersection graphs of finite non-abelian simple groups have an upper bound 28. In particular, the intersection graph of a finite non-abelian simple group is connected.     

Finite dimensional Nichols algebras over certain p-groups

We investigate pointed Hopf algebras over finite nilpotent groups of odd order, with nilpotency class 2. For such a group G, we show that if its commutator subgroup coincides with its center, then there exists no non-trivial finite-dimensional pointed Hopf algebra with kG as its coradical. We apply these results to non-abelian groups of order p³, p⁴ and p⁵, and list all the pointed Hopf algebras of order p⁶, whose group of grouplikes is non-abelian.     

Non-Commuting Graphs of Nilpotent Groups

Let G be a non-abelian group and Z(G) be the center of G. The non-commuting graph Γ _G associated to G is the graph whose vertex set is G?Z(G) and two distinct elements x, y are adjacent if and only if xy ≠ yx. We prove that if G and H are non-abelian nilpotent groups with irregular isomorphic non-commuting graphs, then |G| = |H|.     

On Infinite Camina Groups

A group G is called a Camina group if G′ ≠ G and each element x ∈ G?G′ satisfies the equation x ^G  = xG′, where x ^G denotes the conjugacy class of x in G. Finite Camina groups were introduced by Alan Camina in 1978, and they had been studied since then by many authors. In this article, we start the study of infinite Camina groups. In particular, we characterize infinite Camina groups with a finite G′ (see Theorem 3.1) and we show that infinite non-abelian finitely generated Camina groups must be nonsolvable (see Theorem 4.3). We also describe locally finite Camina groups, residually finite Camina groups (see Section 3) and some periodic solvable Camina groups (see Section 5).     

Finite Groups with Few Non-Abelian Subgroups

Let G be a finite group and τ(G) denote the number of conjugacy classes of all non-abelian subgroups of G. The symbol π(G) denotes the set of the prime divisors of |G|. In this paper, finite groups with τ(G) ≤ |π(G)| are classified completely. Furthermore, finite nonsolvable groups with τ(G) = |π(G)| +1 are determined.     

Localization and finite simple groups

A homomorphism from a groupH to a groupG is a localization if and only if it induces a bijection between Hom(G, G) and Hom(H, G). In this paper we study the equivalence relation that localization induces on the family of finite non-abelian simple groups.     

Local Near-Rings and Triply Factorized Groups

Abstract

Groups G of the form G = AB = AM = BM for two subgroups A and B of G and a normal subgroup M of G with A ∩ M = B ∩ M = 1 are called triply factorized and play an important rôle in the theory of factorized groups. In this paper, a method to construct triply factorized groups with non-abelian M using local near-rings is introduced.     

On central difference sets in certain non-abelian 2-groups

In this note, we define the class of finite groups of Suzuki type, which are non-abelian groups of exponent 4 and class 2 with special properties. A group G of Suzuki type with |G|=2^2s always possesses a non-trivial difference set. We show that if s is odd, G possesses a central difference set, whereas if s is even, G has no non-trivial central difference set.     

A necessary condition for non-abelian finite <Emphasis Type="Italic">p</Emphasis>-groups with second centre of order <Emphasis Type="Italic">p</Emphasis><Superscript>2</Superscript>

Let G be a non-abelian finite p-group such that |Z ₂(G)| = p ². In this paper we prove that each maximal subgroup M ≠ C _G (Z ₂(G)) is non-abelian and has cyclic centre.     

Some remarks on conjugate closed groups

Let G be a group. In this note we define conjugate closed groups, which are briefly called CC — Groups. These groups form a proper subclass of T — Groups. We prove that if G = Z(G) × H, then G is conjugate closed if and only if H is conjugate closed. We also show that a finite group G is semisimple, conjugate closed and perfect if and only if it is a direct product of non-abelian and simple groups.     

Amenability properties of the centres of group algebras

Let G be a locally compact group, and ZL¹(G) be the centre of its group algebra. We show that when G is compact ZL¹(G) is not amenable when G is either non-abelian and connected, or is a product of infinitely many finite non-abelian groups. We also, study, for some non-compact groups G, some conditions which imply amenability and hyper-Tauberian property, for ZL¹(G).     

On Finite p-Groups Whose IA-Automorphisms Are All Class Preserving

Let G be a finite non-abelian p-group, where p is a prime. An automorphism α of G is called a class preserving automorphism if α(x) ∈ x ^G the conjugacy class of x in G, for all x ∈ G. An automorphism α of G is called an IA-automorphism if x ^?1α(x) ∈ G′ for each x ∈ G. In this paper, we give necessary and sufficient conditions on finite p-group G of nilpotency class 2 such that every IA-automorphism is class preserving.     

Non-abelian tensor square of finite-by-nilpotent groups

Let G be a group. We denote by \({\nu(G)}\) an extension of the non-abelian tensor square \({G \otimes G}\) by \({G \times G}\). We prove that if G is finite-by-nilpotent, then the non-abelian tensor square \({G \otimes G}\) is finite-by-nilpotent. Moreover, \({\nu(G)}\) is nilpotent-by-finite (Theorem A). Also we characterize BFC-groups in terms of \({\nu(G)}\) among the groups G in which the derived subgroup is finitely generated (Theorem B).