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

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,p ^a , p ^a n}, then G ≌ G ₀ × H, where H is abelian and G ₀ contains a normal subgroup M × P ₀ of index p. Moreover, M × P ₀ is the set of all elements of G ₀ of conjugacy class sizes p ^a or 1, where M is an abelian π(n)-subgroup of G ₀ and P ₀ is an abelian p-subgroup of G ₀, neither being central in G. Finally, p ^a = p and P/P ₀ acts fixed-point-freely on M and ?(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,p ^a , p ^a q ^b }, where p and q are two distinct primes.