Permutability degrees of finite groups

Abstract

Given a finite group G, we introduce the permutability degree of G, as pd(G) = 1/|G| |L(G)| ?X?L(G)|PG(X)|, where L(G) is the subgroup lattice of G and PG(X) the permutizer of the subgroup X in G, that is, the subgroup generated by all cyclic subgroups of G that permute with X ? L(G). The number pd(G) allows us to find some structural restrictions on G. Successively, we investigate the relations between pd(G), the probability of commuting subgroups sd(G) of G and the probability of commuting elements d(G) of G. Proving some inequalities between pd(G), sd(G) and d(G), we correlate these notions.