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 k₏ (G) +k₏' (G) f (|G|) for all finite groups G. Here k₏ (G) denotes the number of conjugacy classes of nontrivial p-elements in G and k₏' (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 k₏ (G) a and k₏' (G) b with equalities in both cases if and only if G=Cₚ Cb with CG (Cₚ) = Cₚ.