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.

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