Abstract

There are several problems associated with the computation of the conjugacy classes of elements of a finite group, for example, the computation of centralizers and the determination of conjugacy of two elements. The known algorithms for solving these associated problems and the main problem of determining the conjugacy classes are presented. The effectiveness of the algorithms is illustrated by examples. Some applications are briefly discussed.

Introduction

Let G be a finite group. The conjugate, xg, of an element x ∈ G by an element g ∈ G is the element g-1xg. Two elements x and y are conjugate in G if there exists g ∈ G such that xg = y. The relation of being conjugate in G is an equivalence relation on the set of elements of the group G. The equivalence classes are called conjugacy classes of elements. For each element g ∈ G, the map Φg : G → G defined by x ↦ xg is an automorphism of the group G. The maps are called inner automorphisms and they form a normal subgroup of the automorphism group, Aut(G), of G. Therefore, the elements in a conjugacy class are in the same orbit of Aut(G) in its action on G, and have similar properties. Hence, a set of class representatives contains one of each distinct “type” of element in the group.