# characterisitic subgroup implies normal subgroup

• April 8th 2010, 11:03 AM
wutang
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)
• April 8th 2010, 02:24 PM
aliceinwonderland
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.
• April 8th 2010, 04:13 PM
wutang
thanks, that helps a lot