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Math Help - characterisitic subgroup implies normal subgroup

  1. #1
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    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
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  2. #2
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    Quote Originally Posted by wutang View Post
    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 \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.
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  3. #3
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    thanks, that helps a lot
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