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## Main Question or Discussion Point

Let G be a group and H a subgroup of G. We define the following:

[tex]N_{G}(H) = \{g \in G \,\,|\,\, g^{-1}hg \in H,\, for\, all\,\, h\in H\}[/tex]

Show that [tex]N_{G}(H)[/tex] is a subgroup of G.

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I've shown that for all [itex]x,\, y[/itex] of [itex]N_{G}(H)[/itex], [itex]xy[/itex] is an element of [itex]N_{G}(H)[/itex], but how do I show that [itex]x^{-1}[/itex] is an element of [itex]N_{G}(H)[/itex] ?

[tex]N_{G}(H) = \{g \in G \,\,|\,\, g^{-1}hg \in H,\, for\, all\,\, h\in H\}[/tex]

Show that [tex]N_{G}(H)[/tex] is a subgroup of G.

_______________________

I've shown that for all [itex]x,\, y[/itex] of [itex]N_{G}(H)[/itex], [itex]xy[/itex] is an element of [itex]N_{G}(H)[/itex], but how do I show that [itex]x^{-1}[/itex] is an element of [itex]N_{G}(H)[/itex] ?