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

Abstract

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 | G′ | is prime and G′ ≤ Z (G) as well as for groups G which | G′ | is prime and G′ ∩ 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]. © 2007 Elsevier Ltd. All rights reserved.