FINITE GROUPS WITH THREE RELATIVE COMMUTATIVITY DEGREES

For a finite group G and a subgroup H of G, the relative commutativity degree of H in G, denoted by d(H, G), is the probability that an element of H commutes with an element of G. Let D(G) = {d(H,G): H <= G} be the set of all relative commutativity degrees of subgroups of G. It is shown that a finite group G admits three relative commutativity degrees if and only if G/Z(G) is a non-cyclic group of order pq, where p and q are primes. Moreover, we determine all the relative commutativity degrees of some known groups.