Conjugacy Class Sizes Research Articles

Non-solvable groups and the two-prime hypothesis on conjugacy class sizes

For a finite group G, denotes the set of non-trivial conjugacy class sizes of G. The group G is said to satisfy the two-prime hypothesis if, for every a, , the number of primes dividing the greatest common divisor of a and b is at most two. The aim of this article is to investigate the simple groups satisfying the two-prime hypothesis. Also, as a result, some non-solvable groups satisfying the two-prime hypothesis are studied.

The largest conjugacy class size and the nilpotent subgroups of finite groups

Abstract Let H be a nilpotent subgroup of a finite nonabelian group G, let π = π ⁢ ( | H | ) {\pi=\pi(|H|)} and let bcl ⁡ ( G ) {{\operatorname{bcl}}(G)} be the largest conjugacy class size of the group G. In the present paper, we show that | H ⁢ O π ⁢ ( G ) / O π ⁢ ( G ) | < bcl ⁡ ( G ) {|HO_{\pi}(G)/O_{\pi}(G)|<{\operatorname{bcl}}(G)} .

Real class sizes

In this paper, we study the structures of finite groups using some arithmetic conditions on the sizes of real conjugacy classes. We prove that a finite group is solvable if the prime graph on the real class sizes of the group is disconnected. Moreover, we show that if the sizes of all noncentral real conjugacy classes of a finite group G have the same 2-part and the Sylow 2-subgroup of G satisfies a certain condition, then G is solvable.

ON VANISHING CRITERIA THAT CONTROL FINITE GROUP STRUCTURE II

The first author [J. Brough, ‘On vanishing criteria that control finite group structure’, J. Algebra458 (2016), 207–215] has shown that for certain arithmetical results on conjugacy class sizes it is enough to consider only the vanishing conjugacy class sizes. In this paper we further weaken the conditions to consider only vanishing elements of prime power order.

On finite groups with square-free conjugacy class sizes

The first author is supported by Proyecto Prometeo II/2015/011, and the third author acknowledges the doctorate scholarship “Ayudas para la contratacion de personal investigador en formacion de caracter predoctoral granted by Generalitat Valenciana, Spain.

Conjugacy Class Sizes of Elements of Prime-Power Order of Finite Groups

Let G be a finite group and p is a prime. The purpose of this paper is to investigate influences of the sizes of conjugacy classes of \(p^{\prime }\)-element of prime-power order of G on the structure of finite groups. We get some sufficient conditions for a finite group G to be p-nilpotent groups and supersolvable groups. Some known results are generalized.

On the structure of a mutually permutable product of finite groups

Let a finite group \({G = AB}\) be the product of the mutually permutable subgroups A and B. We investigate the structure of G given by conditions on conjugacy class sizes of elements in \({A \cup B}\) . Some recent results are extended.

On p-parts of character degrees and conjugacy class sizes of finite groups

Let G be a finite group and Irr(G) the set of irreducible complex characters of G. Let ep(G) be the largest integer such that pep(G) divides χ(1) for some χ∈Irr(G). We show that |G:F(G)|p≤pkep(G) for a constant k. This settles a conjecture of A. Moretó [13, Conjecture 4]. We also study the related problems of the p-parts of conjugacy class sizes of finite groups.

The largest character degree, conjugacy class size and subgroups of finite groups

ABSTRACTLet G be a finite group, let b(G) denote the largest irreducible character degree of the group G and let bcl(G) denote the largest conjugacy class size of the group G. We study the relations between the sizes of the nilpotent and solvable subgroups of G and b(G). We also study the relations between the sizes of the nilpotent and solvable subgroups of G and bcl(G).

English

For a finite group G denote by N(G) the set of conjugacy class sizes of G. In 1980s, J.G.Thompson posed the following conjecture: If L is a finite nonabelian simple group, G is a finite group with trivial center and N(G) = N(L), then G ≅ L. We prove this conjecture for an infinite class of simple groups. Let p be an odd prime. We show that every finite group G with the property Z(G) = 1 and N(G) = N(A i ) is necessarily isomorphic to A i , where i ∈ {2p, 2p + 1}.

Finite $\smash{p}$\hbox{-}groups of conjugate type {1,p 3}

Abstract We classify finite p-groups, up to isoclinism, which have only two conjugacy class sizes 1 and p 3 {p^{3}} . It turns out that the nilpotency class of such groups is 2.

One-prime power hypothesis for conjugacy class sizes

A finite group G satisfies the one-prime power hypothesis for conjugacy class sizes if any two conjugacy class sizes m and n are either equal or have common divisor a prime power. Taeri conjectured that an insoluble group satisfying this condition is isomorphic to S × A where A is abelian and S ∼= P SL2(q) for q ∈ {4, 8}. We confirm this conjecture.

Thompson’s conjecture on conjugacy class sizes for the simple group PSUn(q)

We show that if [Formula: see text] is a finite centerless group with the same conjugacy class sizes as [Formula: see text], then [Formula: see text] and so verify a conjecture attributed to John G. Thompson.

Structure of finite groups with four conjugacy class sizes of certain elements

ABSTRACTLet G be a finite group. If the set of conjugacy class sizes of primary and biprimary elements is {1,m,n,mk}, where (m,n) = 1 and k > 1 is a proper divisor of n, we prove that G is solvable with an abelian Hall π(m)-subgroup. Moreover, G∕Z(G) is either a Frobenius group or a 2-Frobenius group of order mn. We also give a detailed structure description of G∕Z(G), which is a generalization of [9, Theorem A].

Square-free class sizes in products of groups

We obtain some structural properties of a factorised group G=AB, given that the conjugacy class sizes of certain elements in A∪B are not divisible by p2, for some prime p. The case when G=AB is a mutually permutable product is especially considered.

The Influence of Conjugacy Class Sizes on the Structure of Finite Groups

Let $G$ be a group. The question of how certain arithmetical conditions on the sizes of the conjugacy classes of $G$ influence the group structure has been studied by many authors. In this paper, we investigate the influence of conjugacy class sizes of primary and biprimary elements on the structure of $G$. A criterion for a group to have abelian Sylow subgroups is given and some well-known results on Baer-groups are generalized.

New characterization of symmetric groups of prime degree

Abstract In this paper, will show that a symmetric group of prime degree p ≥ 5 is recognizable by its order and a special conjugacy class size of (p − 1)!.

The Divisibility Graph of finite groups of Lie type

The Divisibility Graph of a finite group G has vertex set the set of conjugacy class sizes of non-central elements in G and two vertices are connected by an edge if one divides the other. We determine the connected components of the Divisibility Graph of the finite groups of Lie type in odd characteristic.

Frobenius Groups with Almost Distinct Conjugacy Class Sizes

Let G be a finite Frobenius group with at most two conjugacy classes of each size. In this paper, we shall prove that G is isomorphic to \(S_3,\, D_{10},\, A_4,\, \mathbb {Z}_7\rtimes \mathbb {Z}_3,\text { or } (\mathbb {Z}_{5}\times \mathbb {Z}_5)\rtimes (\mathbb {Z}_3\rtimes \mathbb {Z}_4)\).

On normal subgroups with consecutive G-class sizes

Let [Formula: see text] be a group and [Formula: see text] be a normal subgroup of [Formula: see text]. If the set [Formula: see text] is composed by consecutive integers, then [Formula: see text] is either nilpotent or a quasi-Frobenius group with abelian kernel and complements. This is a generalization of Theorem 2 of [A. Beltrán, M. J. Felipe and C. G. Shao, [Formula: see text]-divisibility of conjugacy class sizes and normal [Formula: see text]-complements, J. Group Theory 18 (2015) 133–141].