Remark

(1) Let G be a finite group and $p$ a prime number. By no. 6, Theorem 3 (a) and no. 6, Theorem 2, the following conditions are equivalent: (i) there exists a normal Sylow $p$-subgroup of G; (ii) every Sylow $p$-subgroup of G is normal; (iii) there exists only one Sylow $p$-subgroup of G.

(2) Let G be a nilpotent finite group. Let I be the set of prime divisors of Card(G). By Theorem 4 and Remark 1, $G = \prod_{p \in I} G_p$, where $G_p$ is the unique Sylow $p$-group of G.

(3) Applied to commutative groups, Theorem 4 gives the decomposition, of commutative finite groups as a product of primary components, which will be studied from another point of view in Chapter VII.