More on Landau's theorem and Conjugacy Classes

Let p be a prime. We construct a function f on the natural numbers such that f(x) → ∞ as x → ∞ and kₚ(G)+kp'(G)≥ f(|G|) for all finite groups G. Here kₚ(G) denotes the number of conjugacy classes of nontrivial p-elements in G and kp'(G) denotes the number of conjugacy classes of elements of G whose orders are coprime to p. This is a variation of an old theorem of Landau and is used to prove the following: There exists a number c such that whenever p is a prime and G is a finite group of order divisible by p with $|G|>c$, there exists a factorization $p-1 = ab$ with a and b positive integers such that kₚ(G) ≥ a and kp'(G) ≥ b with equalities in both cases if and only if G=Cₚ Cb with CG(Cₚ) = Cₚ.