Some considerations on the n-th commutativity degrees of finite groups

Let G be a finite group and n a positive integer. The n-th commutativity degree P-n(G) of G is the probability that the n-th power of a random element of G commutes with another random element of G. In 1968, P. Erdos and P.Turan investigated the case n = 1, involving only methods of combinatorics. Later several authors improved their studies and there is a growing literature on the topic in the last 10 years. We introduce the relative n-th commutativity degree P-n(H, G) of a subgroup H of G. This is the probability that an n-th power of a random element in H commutes with an element in G. The influence of P, (G) and P-n (H, G) on the structure of G is the purpose of the present work.