  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
$C_G(C_p) = C_p$.
