An extension of a theorem of Alan Camina on conjugacy class sizes

Abstract

Let G be a finite group. Let n be a positive integer and p a prime coprime to n. In this paper we prove that if the set of conjugacy class sizes of primary and biprimary elements of group G is {1,pa, pan}, then G ≌ G0 × H, where H is abelian and G0 contains a normal subgroup M × P0 of index p. Moreover, M × P0 is the set of all elements of G0 of conjugacy class sizes pa or 1, where M is an abelian π(n)-subgroup of G0 and P0 is an abelian p-subgroup of G0, neither being central in G. Finally, pa = p and P/P0 acts fixed-point-freely on M and ϕ(P) ≤ Z(P). This is an extension of Alan Camina’s theorems on the structure of groups whose set of conjugacy class size is {1,pa, paqb}, where p and q are two distinct primes.