characterisitic subgroup implies normal subgroup

Recall that a subgroup N of a group G is called characteristic if F(N)=N for all automorphisms F of G. If N is a characteristic subroup of G, show that N is a normal subgroup of G.
I don't have a clue were to start this from, any help would be great(Rofl)

Quote:

Originally Posted by wutang

Recall that a subgroup N of a group G is called characteristic if F(N)=N for all automorphisms F of G. If N is a characteristic subroup of G, show that N is a normal subgroup of G.
I don't have a clue were to start this from, any help would be great(Rofl)

If F(N)=N for all automorphisms F of G, then $\phi_g(N) = N$ for every inner automorphism $\phi_g \in Inn(G) \subset Aut(G)$ .
Now $\phi_g(N)=N$ for every $\phi_g \in Inn(G)$ implies that $gNg^{-1}=N$ for every $g \in G$ . It follows that N is a normal subgroup of G.

thanks, that helps a lot