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 nite group G satises [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 nite groups, Bull. Belgian Math. Soc. 19 (2012), 373-382]. Numerical inequalities will be illustrated, in order to nd connections with the subgroup commutativity degree and with the commutativity degree of G. On the other hand, the literature has known a signicant 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.

Mathematics Subject Classication (2010): 20D60, 06B23, 20P05.
Key words: Inequalities, centralizers, probability, measure, dihedral groups.