ABSTRACT In this paper we present two new results on the number of certain conjugacy classes of a finite group. For a finite group G, let n (G) be the maximum of k (G) taken over all primes p where k (G) denotes the number of conjugacy classes of nontrivial p-elements in G. Using a recent theorem of Giudici, Morgan and Praeger, we prove that there exists a function f (x) with f (x) as x such that n (G) f (|G|) for any finite group G. Let G be a finite group, and let p be a prime dividing |G|. Let k^ (G) denote the number of conjugacy classes of elements of G whose orders are coprime to p. We show that either p=11 and G=C₁₁² SL (2, 5), or there exists a factorization p-1 = ab with a and b positive integers, such that k (G) a and k^ (G) b with equalities in both cases if and only if G=Cₚ Cb with CG (Cₚ) = Cₚ.
Çınarcı et al. (Sat,) studied this question.