More on Landau's theorem and Conjugacy Classes

Let \(p\) be a prime. We construct a function \(f\) on the natural numbers such that \(f(x) \to \infty\) as \(x \to \infty\) and \(k_{p}(G)+k_{p'}(G)\geq f(|G|)\) for all finite groups \(G\). Here \(k_{p}(G)\) denotes the number of conjugacy classes of nontrivial \(p\)-elements in \(G\) and \(k_{p'}(G)\) 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 \(k_{p}(G) \geq a\) and \(k_{p'}(G) \geq b\) with equalities in both cases if and only if \(G=C_p \rtimes C_b\) with \(C_G(C_p) = C_p\).