Moderate# Characterizing the Normalizer of Subgroups

ABSALG-R1J9ZB

Recall that for every subgroup $H$ of a group $G$, the normalizer of $H$ in $G$ is denoted by $N_G(H)$, or simply $N(H)$ and is defined as $N(H)=\{g\in G:gHg^{-1}=H\}$.

Which of the following facts is true about the normalizer?