Exercise 28, § 6

Let G be a finite group and $p$ a prime number. An element $s \in G$ is called p-unipotent (resp. p-regular) if its order is a power of $p$ (resp. not divisible by $p$).

(a) Let $x \in G$. Show that there exists a unique ordered 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 Sylow $p$-group of G, C its centralizer and E the set of p-regular elements of G. Show that

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