## A criterion for normalcy of finite subgroups

Let $G$ be a group and $N$ a finite subgroup of $G$. Show that $gNg^{-1} \subseteq N$ if and only if $gNg^{-1}$. Deduce that $N_G(N) = \{ g \in G \ |\ gNg^{-1} \subseteq N \}$.

It suffices to show that for all $g \in G$, $gNg^{-1} \subseteq N$ implies $gNg^{-1} = N$. Let $g \in G$. The mapping $n \mapsto gng^{-1}$ is a bijection $N \rightarrow gNg^{-1}$, so that $|gNg^{-1}| = |N|$. Since $N$ is finite, $gNg^{-1} = N$.

The last statement follows trivially.