Theorem 14.13

Statement

The following are three equivalent conditions for a subgroup $H$ of a group $G$ to be a normal subgroup of $G$.

1. $ghg^{-1} \in H$ for all $g \in G$ and $h \in H$.
2. $gHg^{-1} = H$ for all $g \in G$.
3. $gH = Hg$ for all $g \in G$.

Condition (2) of this theorem is often taken as the definition of a normal subgroup $H$ of a group $G$.

Proof

NA