Groups with few conjugacy classes (original) (raw)

Conjugacy deficiency in finite groups

2013

We consider the function r(G) = |G|−k(G), where the group G has exactly k(G) conjugacy classes. We find all G where r(G) is small and pose a number of relevant questions.

On powers of conjugacy classes in finite groups

Journal of Group Theory, 2022

Let 𝐾 and 𝐷 be conjugacy classes of a finite group 𝐺, and suppose that we have K n = D ∪ D - 1 K^{n}=D\cup D^{-1} for some integer n ≥ 2 n\geq 2 . Under these assumptions, it was conjectured that ⟨ K ⟩ \langle K\rangle must be a (normal) solvable subgroup of 𝐺. Recently R. D. Camina has demonstrated that the conjecture is valid for any n ≥ 4 n\geq 4 , and this is done by applying combinatorial results, the main of which concerns subsets with small doubling in a finite group. In this note, we solve the case n = 3 n=3 by appealing to other combinatorial results, such as an estimate of the cardinality of the product of two normal sets in a finite group as well as to some recent techniques and theorems.

Two problems on finite groups with k conjugate classes

Journal of the Australian Mathematical Society, 1968

Let G be a finite group of order g having exactly k conjugate classes. Let π(G) denote the set of prime divisors of g. K. A. Hirsch [4] has shown that By the same methods we prove g ≡ k modulo G.C.D. {(p–1)2 p ∈ π(G)} and that if G is a p-group, g = h modulo (p−1)(p2−1). It follows that k has the form (n+r(p−1)) (p2−1)+pe where r and n are integers ≧ 0, p is a prime, e is 0 or 1, and g = p2n+e. This has been established using representation theory by Philip Hall [3] (see also [5]). If then simple examples show (for 6 ∤ g obviously) that g ≡ k modulo σ or even σ/2 is not generally true.

Groups with Restricted Conjugacy Classes

2002

Let F C0 be the class of all finite groups, and for each non- negative integer n define by induction the group class F Cn+1 consisting of all groups G such that for every element x the factor group G/CG(h xi G) has the property F Cn. Thus F C1-groups are precisely groups with finite conjugacy classes, and the class F Cn obviously contains all finite groups and all nilpotent groups with class at most n. In this paper the known theory of F C-groups is taken as a model, and it is shown that many properties of F C-groups have an analogue in the class of F Cn-groups.

Some Problems About Products of Conjugacy Classes in Finite Groups

International Journal of Group Theory, 2019

‎We summarize several results about non-simplicity‎, ‎solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes‎. ‎We also collect some problems that have only been partially solved‎.

Finite groups with a given number of conjugate classes

Canadian Journal of Mathematics, 1968

This paper presents a list of all finite groups having exactly six and seven conjugate classes and an outline of the background necessary for the proof, and gives, in particular, two results which may be of independent interest. In 1903 E. Landau (8) proved, by induction, that for each the equation * has only finitely many solutions over the positive integers.