On the commuting probability in finite groups

For Bernd Fischer on the occasion of his 70th birthday Introduction: When G is a finite group, we may endow G×G with the structure of a probability space by assigning the uniform distribution. As was pointed out by W.H. Gustafson [10], the probability that a randomly chosen pair of elements of G commute is then k(G)|G | , where k(G) is the number of conjugacy classes of G. We denote this probability by cp(G). It was also noted in [10] that cp(G) ≤ 58 fo