Proposition 12

Let G be a p-group and H a subgroup of G distinct from G. Then:

(a) The normalizer $N_G(H)$ of H in G is distinct from G. (b) There exists a normal subgroup N of G of index p in G, which contains H.

Assertion (a) follows from no. 3, Corollary 1 to Proposition 8. We prove (b). By no. 3, Corollary 2 to Proposition 8, there exists a normal subgroup $N'$ of G containing H, distinct from G and such that $G/N'$ is commutative. Let N be a maximal subgroup distinct from G containing $N'$. Then N is normal (no. 2, Corollary 3 to Proposition 6) and $G/N$ is a simple commutative $p$-group and hence cyclic of order $p$ (§ 4, no. 10, Corollary to Proposition 20).