Commutativity degrees and related invariants of some finite nilpotent groups

Abstract

In this research, two-generator p-groups of nilpotency class two, which is referred to as G are considered. The commutativity degree of a finite group G, denoted as P?G?, is defined as the probability that a random element of the group G commutes with another random element in G. The main objective of this research is to derive the general formula for P?G? and its generalizations. This research starts by finding the formula for the number of conjugacy classes of G. Then the commutativity degree of each of these groups is determined by using the fact that the commutativity degree of a finite group G is equal to the number of conjugacy classes of G divided by the order of G. The commutativity degree can be generalized to the concepts of n-th commutativity degree, ? ?, n P G which is defined as the probability that the n-th power of a random element commutes with another random element from the same group. Moreover, ? ? n P G can be extended to the relative n-th commutativity degree, ? , ?, n P H G which is the probability of commuting the n-th power of a random element of H with an element of G, where H is a subgroup of G. In this research, the explicit formulas for ? ? n P G and ? , ? n P H G are computed. Meanwhile, another generalization of the commutativity degree, which is called commutator degree and denoted by ? ? g P G , is the probability that the commutator of two elements in G is equal to an element g in G. In this research, an effective character-free method is used for finding the exact formula for ? ?. g P G Finally, the exterior degree of the wreath product of A and B, P^ ?A? B?, is found where A and B are two finite abelian groups.