Related papers: Centralizers and conjugacy classes in finite class…

We provide two alternative ways to determine the number of (bi-)twisted conjugacy classes in a finite group: one by counting certain irreducible characters and one by counting certain twisted conjugacy classes of other endomorphisms. In…

Finite group theorists have established many formulas that express interesting properties of a finite group in terms of sums of characters of the group. An obstacle to applying these formulas is lack of control over the dimensions of…

The action of any group on itself by conjugation and the corresponding conjugacy relation play an important role in group theory. There have been several attempts to extend the notion of conjugacy to semigroups. In this paper, we present a…

We answer two questions on the complexities of decision problems of groups, each related to a classical result. First, C. Miller characterized the complexity of the isomorphism problem for finitely presented groups in 1971. We do the same…

The derivation of the brackets among coordinates and momenta for classical constrained systems is a necessary step toward their quantization. Here we present a new approach for the determination of the classical brackets which does neither…

We classify pairs of conjugacy classes in almost simple algebraic groups whose product consists of finitely many classes. This leads to several interesting families of examples which are related to a generalization of the Baer--Suzuki…

Let C be the centralizer in a finite Weyl group of an elementary abelian 2-subgroup. We show that every complex representation of C can be realized over the field of rational numbers. The same holds for a Sylow 2-subgroup of C.

A collection C of subgroups of a finite group G can give rise to three different standard formulas for the cohomology of G in terms of either: the subgroups in C; or their centralizers; or their normalizers. We give a short but systematic…

In this paper we give an algorithm to determine all finite matrix groups over a number field. Our algorithm is based on the representation theory of finite groups.

The problem of finding a canonical form of complex matrices up to conjugacy with the set of canonical matrices being a union of affine planes in the matrix space is considered. A solution of the problem is given producing a new canonical…

The role of finite centralizers of involutions in pseudo-finite groups is analyzed. It is shown that a pseudo-finite group admitting a definable involutory automorphism fixing only finitely many elements is finite-by-abelian-by-finite. As a…

We consider the following two finiteness conditions on normalizers and centralizers in a group G: (i) |N_G(H):H| is finite for every non-normal subgroup H of G, and (ii) |C_G(x):<x>| is finite for every non-normal cyclic subgroup <x> of G.…

Landau's theorem on conjugacy classes asserts that there are only finitely many finite groups, up to isomorphism, with exactly $k$ conjugacy classes for any positive integer $k$. We show that, for any positive integers $n$ and $s$, there…

In the present paper we discuss some recent versions of localisation methods for calculations in the groups of points of algebraic-like and classical-like groups. Namely, we describe relative localisation, universal localisation, and…

Let $G$ be a finite $p$-group such that $x\Z(G) \subseteq x^G$ for all $x \in G- \Z(G)$, where $x^G$ denotes the conjugacy class of $x$ in $G$. Then $|G|$ divides $|\Aut(G)|$, where $\Aut(G)$ is the group of all automorphisms of $G$.

Finite groups with given systems of permuteral and strongly permuteral subgroups are studied. New characterizations of w-supersoluble and supersoluble groups are received.

In this paper, we count the number of conjugacy and automorphism-conjugacy classes of elements of a ZM-group. The size of a conjugacy class with respect to these two equivalence relations is also counted.

We compute conjugacy classes in maximal parabolic subgroups of the general linear group. This computation proceeds by reducing to a ``matrix problem''. Such problems involve finding normal forms for matrices under a specified set of row and…

We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…

Let $G$ be a finite group and let $p$ be a prime. In this paper, we study the structure of finite groups with a large number of $p$-regular conjugacy classes or, equivalently, a large number of irreducible $p$-modular representations. We…