We discuss the time complexity of the word and conjugacy problems for free products G = A ?C B of groups A and B with amalgamation over a subgroup C. We stratify the set of elements of G with respect to the complexity of the word and conjugacy problems and show that for the generic stratum the conjugacy search problem is decidable under some reasonable assumptions about groups A, B, C. Moreover...

Throughout this paper G is a finite group and S is a subgroup of G. When S is contained in the subgroup H of G, we say S is strongly closed in H with respect to G if whenever s ∈ S and g ∈ G are such that s ∈ H, then s ∈ S. In other words, every G-conjugacy class of elements of S intersected with H is contained in S. In the case where S is a p-group for some prime p we say that S is strongly cl...

Let G be a finite group, k a commutative ring upon which G acts. For every subgroup H of G, the trace (or norm) map trH : k → k H is defined. trH is onto if and only if there exists an element xH such that trH(xH) = 1. We will show that the existence of xP for every subgroup P of prime order determines the existence of xG by exhibiting an explicit formula for xG in terms of the xP , where P var...

The commuting graph of a group is a graph with vertexes set of a subset of a group and two element are adjacent if they commute. The aim of this paper is to obtain the automorphism group of the commuting graph of a conjugacy class in the symmetric groups. The clique number, coloring number, independent number, and diameter of these graphs are also computed.

We discuss a Plancherel formula for countable groups, which provides canonical decomposition of the regular representation such group Γ into direct integral factor representations. Our main result gives precise description this in terms FC-center Γfc (that is, normal subgroup consisting elements with finite conjugacy class); involves action an appropriate totally disconnected compact automorphi...

Given a group $G$, we write $x^G$ for the conjugacy class of $G$ containing element $x$. A famous theorem B. H. Neumann states that if is in which all classes are finite with bounded size, then derived $G'$ finite. We establish following result. Let $n$ be positive integer and $K$ subgroup such $|x^G|\leq n$ each $x\in K$. $H=\langle K^G\rangle$ normal closure $K$. Then order $H'$ $n$-bounded. ...

Let G be a finite group and S proper subgroup of G. A is called an S-(CAP)-group if every local either CAP-subgroup or conjugate to
 S. The main purpose this construction to demonstrate new way analyzing the structure by properties number conjugacy classes its subgroups.

Recently, Macdonald et. al. showed that many algorithmic problems for finitely generated nilpotent groups including computation of normal forms, the subgroup membership problem, the conjugacy problem, and computation of subgroup presentations can be done in LOGSPACE. Here we follow their approach and show that all these problems are complete for the uniform circuit class TC0 – uniformly for all...

Given a short exact sequence of groups with certain conditions, 1 → F → G → H → 1, we prove that G has solvable conjugacy problem if and only if the corresponding action subgroup A Aut(F ) is orbit decidable. From this, we deduce that the conjugacy problem is solvable, among others, for all groups of the form Z2 Fm, F2 Fm, Fn Z, and Zn A Fm with virtually solvable action group A GLn(Z). Also, w...