Probability that the commutator of two group elements is equal to a given element

摘要

In this paper we study the probability that the commutator of two randomly chosen elements in a finite group is equal to a given element of that group. Explicit computations are obtained for groups G which vertical bar G'vertical bar is prime and G' <= Z(G) as well as for groups G which vertical bar G'vertical bar is prime and G' boolean AND Z(G) = 1. This paper extends results of Rusin [see D.J. Rusin, What is the probability that two elements of a finite group commute? Pacific J. Math. 82 (1) (1979) 237-247]. (c) 2007 Elsevier B.V. All rights reserved.