In this paper we establish decomposition theorems for derivations of group rings. We provide a topological technique studying ring A[G] in case G has finite conjugacy classes. As result, describe all algebra the when is group, or an FC-group. addition, algorithm to explicitly calculate finite. examples, Z4[S3] and F2mD2n are considered.

In 2000, L. Héthelyi and B. Külshammer proved that if p is a prime number dividing the order of a finite solvable group G, then G has at least 2 √ p − 1 conjugacy classes. In this paper we show that if p is large, the result remains true for arbitrary finite groups.

1 2 3 Let T be a rooted tree and Iso(T ) be the group of its isometries. We study closed subgroups G of Iso(T ) with respect to the number of conjugacy classes of Iso(T ) having representatives in G.

Abstract We prove that there exists a universal constant D such if p is prime divisor of the index Fitting subgroup finite group G , then number conjugacy classes at least $Dp/\log_2p$ . conjecture we can take $D=1$ and for solvable groups, $D=1/3$

In this work, we study direct limits of finite dimensional basic classical simple Lie superalgebras and obtain the conjugacy classes of Cartan subalgebras under the group of automorphisms.

Let G be a group. Let π(G) be the set of prime divisor of |G|. Let GK(G) denote the graph with vertex set π(G) such that two primes p and q in π(G) are joined by an edge if G has an element of order p · q. We use s(G) to denote the number of connected components of the prime graph GK(G). Let N(G) be the set of nonidentity orders of conjugacy classes of elements in G. Some authors have proved th...

For a given finite group G, we have that the number of irreducible representations of G is equal to the number of conjugacy classes of G. Although numerically equal, there is no general formula to map the conjugacy classes of a group to its irreducible representations. For the case of the symmetric group Sn, however, there is a remarkably simple correspondence; we will see that each irreducible...

Motivated by a question of Gordon and Wilton, we consider the question of which collections of words are “virtually geometric”. In particular, we prove that some words (e.g. bbaaccabc) are not virtually geometric. 1. Definitions and introduction Let G be a group, g = {[gi]}i∈I a collection of conjugacy classes in G, and X some space. Any homomorphism φ : G → π1(X) induces a map, φg : ⊔

Let G be a pro-p group of finite cohomological dimension and type FP∞ and T is a finite p-group of automorphisms of G. We prove that the group of fixed points of T in G is again a pro-p group of type FP∞ (in particular it is finitely presented). Moreover we prove that a pro-p group G of type FP∞ and finite virtual cohomological dimension has finitely many conjugacy classes of finite subgroups.

Random walk on the chambers of hyperplanes arrangements is used to de ne a family of card shu ing measuresMW;x for a nite Coxeter group W and real x 6= 0. By algebraic group theory, there is a map from the semisimple orbits of the adjoint action of a nite group of Lie type on its Lie algebra to the conjugacy classes of the Weyl group. Choosing such a semisimple orbit uniformly at random thereby...