More on landau’s theorem and conjugacy classes

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 kₚ(G) taken over all primes p where kₚ(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_p(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 kₚ(G) ≥ a and k_p(G) ≥ b with equalities in both cases if and only if G=Cₚ Cb with CG(Cₚ) = Cₚ.