Theorem 4

Let $G$ be a finite group. The following conditions are equivalent:

(a) $G$ is nilpotent. (b) $G$ is a product of $p$-groups. (c) For every prime number $p$ there exists a normal Sylow $p$-subgroup of $G$. (b) $\Rightarrow$ (a) (no. 5, Corollary to Theorem 1).

Suppose (a) holds and let $P$ be a Sylow $p$-subgroup of $G$. If $N$ is the normalizer of $P$ in $G$, Corollary 1 to Theorem 3 shows that $N$ is its own normalizer. By § 6, no. 3, Corollary to Proposition 8, this shows that $N = G$. Hence (a) $\Rightarrow$ (c).

Suppose (c) holds and let $I$ be the set of prime numbers dividing $\mathrm{Card}(G)$. For all $p \in I$, let $P_p$ be a normal Sylow $p$-subgroup of $G$. For all $p \neq q$, $P_p \cap P_q$ is reduced to $e$ for it is both a $p$-group and a $q$-group, hence $P_p$ and $P_q$ centralize one another ($§ 4$, no. 9, Proposition 15). Let $\phi$ be the canonical homomorphism ($§ 4$, no. 9, Proposition 12) of $\prod_{p \in I} P_p$ into $G$. The homomorphism $\phi$ is surjective by the Remark of no. 6. As $\mathrm{Card}\left(\prod_{p \in I} P_p\right) = \mathrm{Card}(G)$, it follows that $\phi$ is bijective.