# Thread: characterisitic subgroup implies normal subgroup

1. ## 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

2. 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
If F(N)=N for all automorphisms F of G, then $\displaystyle \phi_g(N) = N$ for every inner automorphism $\displaystyle \phi_g \in Inn(G) \subset Aut(G)$.
Now $\displaystyle \phi_g(N)=N$ for every $\displaystyle \phi_g \in Inn(G)$ implies that $\displaystyle gNg^{-1}=N$ for every $\displaystyle g \in G$. It follows that N is a normal subgroup of G.

3. thanks, that helps a lot