Corollary 1

Tag 05DF, permanent. Cite it as [Bourbaki, Tag 05DF].

Let $G$ be a nilpotent group and $H$ a subgroup of $G$ . If $H$ is distinct from $G$ , the normalizer $N_G(H)$ of $H$ in $G$ is distinct from $H$ .

Let $k$ be the largest index such that $H^k \neq H$ . The group $H^k$ normalizes $H$ and is distinct from $H$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.