Number Of Conjugacy Classes Research Articles

A density theorem on the number of conjugacy classes in finite groups

For each finite group G with k(G) conjugacy classes and order g, it is well known that g (log n)κ

Finite groups with few non-linear irreducible characters

If we denote the number of conjugate classes of a finite group G by k, then we have k>=[G:G'I and the equality sign holds if and only if G is abelian. In fact, if two elements of G are conjugate then they are obviously in the same coset of G' and G' itself consists of at least two conjugate classes unless G'= 1. The non-abelian groups which have in a sense the smallest possible number of conjugate classes are those with k= [G: G'I+ 1. Our aim in this paper is to give a full description of these groups. These are at the same time the non-abelian groups whose character tables can be constructed the most easily, as they have exactly one non-linear irreducible chat'acter. We also show that if G has at most 3 non-linear irreducible characters then G is solvable. This result cannot be improved as A s has 4 non-linear irreducible characters. As the method of proof will be purely group-theoretic we prefer the formulation of the theorems given below to the one in terms of the number of non-linear irreducible characters.

The Fitting Length of a Finite Soluble Group and the Number of Conjugacy Classes of its Maximal Metanilpotent Subgroups

It is known that the Fitting length h(G) of a finite soluble group G is bounded in terms of the number v(G) of the conjugacy classes of its maximal nilpotent subgroups. For |G| odd, a bound on h(G) in terms of v(G) was discussed in Lausch and Makan [6]. In the case when the prime 2 divides |G|, a logarithmic bound on h(G) in terms of v(G) is obtained in [7]. The main purpose of this paper is to show that the Fitting length of a finite soluble group is also bounded in terms of the number of conjugacy classes of its maximal metanilpotent subgroups. In fact, our result is rather more general.

On the radical of the group algebra of a 𝑝-group over a modular field

Let G be a finite p-group, K be the field of integers modulo p, KG be the group algebra of G over K and N be the radical of KG. By using the fact that the annihilator, A ( N ) A(N) , of N is one dimensional, we characterize the elements of A ( N 2 ) A({N^2}) . We also present relationships among the cardinality of A ( N 2 ) A({N^2}) , the number of maximal subgroups in G and the number of conjugate classes in G. Theorems concerning the Frattini subalgebra of N and the existence of an outer automorphism of N are also proved.

The number of conjugacy classes in a finite group

The number of conjugacy classes in a finite group

Absolutely faithful group actions

The definition and main result. It has been shown ((1), § 3) that if G is any finite group and p any prime number not dividing |G|, then the number of conjugacy classes of maximal nilpotent subgroups in the regular wreath product of a cyclic group of order p by G is equal to the number of conjugacy classes of all nilpotent subgroups in G. This fact, together with various properties of the map by means of which it was established, proved helpful in dealing with questions of construction raised in (1). The present note isolates the key property of the wreath product on which the argument rests, and from this shows how the argument can be carried over to a more general context. The essential situation is that a group G acts on a group A in a way which will be called ‘absolutely faithful’.

Finite Groups with a given Number of Conjugate Classes

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.

Orbits of Torus Groups Operating on Manifolds

Suppose that G is a compact connected Lie group operating on a compact manifold X. If x E X, the isotropy group Gx at x is the set of g E G which have x fixed. Montgomery [3] has proposed the following basic problem: are there only a finite number of conjugate classes of Gx's*? Yang [11] has answered this question in the affirmative for the case in which G operates differentiably on X. We show here that the answer is also in the affirmative when G is a torus group operating (not necessarily differentiably) on a compact orientable manifold. Mostow [7], assuming the above result on torus groups, has showed that the answer to Montgomery's question is affirmative in the general case. As a basic procedure, we use the existence in G of a plentiful supply of subgroups Gi of prime-power order. We apply the Smith special homology theory [8, 9] for transformation groups of this type to show that if Gi is a sequence of such subgroups of G with lim Gi = G then the fixed point sets F(G0) of Gi converge regularly in an appropriate sense of homology to the fixed point set FG Of G. Using the theory of regular convergence and the Smith theory, we conclude that FGi = FG for i large. The main theorem follows readily from this fact. We assume a knowledge of the Smith homology theory [8, 9].