Theorem 14.13

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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 $G$.



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