Exercise 28, § 6

Let G be a finite group and let $p$ be a prime number. An element $s \in G$ is said to be $p$-unipotent (resp. $p$-regular) if its order is a power of $p$ (resp. is not divisible by $p$). a) Let $x \in G$. Now prove that there exists a unique pair $(u, t)$ of elements of G satisfying the following conditions: $u$ is $p$-unipotent, $t$ is $p$-regular, $x = ut = tu$. (Consider first the case where G is the cyclic group generated by x.) b) Let P be a $p$-Sylow group of G, let C be its centralizer and let E be the set of $p$-regular elements of G. Prove that

Deduce that $\operatorname{Card}(E) \not\equiv 0 \pmod{p}$. (Reason by induction on $\operatorname{Card}(G)$ and reduce to the case where $C = G$; then use a) to prove that $\operatorname{Card}(E) = (G : P)$.)