We started our research with the intent on answering the following question: can we find a way to calculate all the subgroups of the symmetric group. This is easier said that done, as the number of subgroups for a symmetric group grows quickly with each successive symmetric group. This problem can actually be simplified to finding the subgroup conjugacy classes. So now we have the slightly diff...

Let H,,(q) be the Iwahori-Hecke algebra of the symmetric group S,,. Let F, be a finite field with y elements and let B be the Bore1 subgroup of upper triangular matrices in the general linear group G = GL,(F,). Let 1: denote the trivial representation of B induced to G. Then H,,(q) has a natural action on 1; that commutes with the G-action, and we define the bitrace btr(g, a) to be the trace of...

Let G be a connected, reductive group over an algebraically closed field of good characteristic. For u ∈ G unipotent, we describe the conjugacy classes in the component group A(u) of the centralizer of u. Our results extend work of the second author done for simple, adjoint G over the complex numbers. When G is simple and adjoint, the previous work of the second author makes our description com...

Let G be a simple algebraic group over an algebraically closed field k of bad characteristic. We classify the spherical unipotent conjugacy classes of G. We also show that if the characteristic of k is 2, then the fixed point subgroup of every involutorial automorphism (involution) of G is a spherical subgroup of G.

We define and study a correspondence between the set of distinguished G0-conjugacy classes in a fixed connected component of a reductive group G (with G0 almost simple) and the set of (twisted) elliptic conjugacy classes in the Weyl group. We also prove a homogeneity property related to this correspondence. Introduction 0.1. Let k be an algebraically closed field of characteristic p ≥ 0 and let...

It is observed that the conjugacy growth series of the infinite finitary symmetric group with respect to the generating set of transpositions is the generating series of the partition function. Other conjugacy growth series are computed, for other generating sets, for restricted permutational wreath products of finite groups by the finitary symmetric group, and for alternating groups. Similar m...

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.

Let Uε(g) be the simply connected quantized enveloping algebra associated to an exceptional simple Lie algebra g at the roots of unity. The De Concini-Kac-Procesi conjecture on the dimension of the irreducible representations of Uε(g) is proved for representations corresponding to the spherical conjugacy classes of the simply connected group with Lie algebra g.

A class function φ on a finite group G is said to be an order separator if, for every x and y in G\{1} , φ(x) = φ(y) is equivalent to x and y being of the same order. Similarly, φ is said to be a class-size separator if, for every x and y in G \ {1} , φ(x) = φ(y) is equivalent to |CG(x)| = |CG(y)| . In this paper, finite groups whose nonlinear irreducible complex characters are all order separa...

By algebraic group theory, there is a map from the semisimple conjugacy classes of a nite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on conjugacy classes of the Weyl group. We conjecture that this measure agrees with a second measure on conjugacy classes of the Weyl group induced by a construction of ...