Strong subgroup commutativity degree and some recent problems on the commuting probabilities of elements and subgroups

Abstract

We show some results on the probability that a randomly picked pair (H, K) of subgroups of a finite group G satisfies [H, K] = 1. This notion of probability is related with the subgroup S-commutativity degree in [D.E. Otera and F.G. Russo, Subgroup S-commutativity degree of finite groups, Bull. Belgian Math. Soc. 19 (2012), 373–382]. Numerical inequalities will be illustrated, in order to find connections with the subgroup commutativity degree and with the commutativity degree of G. On the other hand, the literature has known a significant growth in the last years and so we discuss a series of new open problems, which are arousing interest in the area. The presence of a large bibliography allows us to have a detailed overview of the status of knowledge on the topic.