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Math Help - Characteristic normal subgroups

  1. #1
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    Characteristic normal subgroups

    Prove that every characteristic subgroup of a normal subgroup of a group G is a normal subgroup of G, and that every characteristic subgroup of a characteristic subgroup of a group G is a characteristic subgroup of G.

    Proof so far:

    Suppose that N is a normal subgroup of G, and let H be a charcteristic subgroup of N. Find an inner automorphism of H such that  f:H \mapsto H by f(h) = nhn^{-1} \ \ \ n \in N with  f(H) = H

    Now, since N is normal, we have gNg^{-1} \in N, so we have  gHg^{-1} = g f(H) g^{-1} = g ( nhn^{-1} ) g^{-1} = (gn)h(gn)^{-1}

    So H is normal in G.

    Is this right?
    Last edited by ThePerfectHacker; September 25th 2008 at 07:52 AM.
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Proof so far:

    Suppose that N is a normal subgroup of G, and let H be a charcteristic subgroup of N. Find an inner automorphism of H such that  f:H \mapsto H by f(h) = nhn^{-1} \ \ \ n \in N with  f(H) = H

    Now, since N is normal, we have gNg^{-1} \in N, so we have  gHg^{-1} = g f(H) g^{-1} = g ( nhn^{-1} ) g^{-1} = (gn)h(gn)^{-1}

    So H is normal in G.

    Is this right?
    What you wrote is messy. Let H be a charachteristic subgroup. And let g\in G. We want to show ghg^{-1} \in H for h\in H. Define \theta: N\to N by \theta (n) = gng^{-1} - this is an automorphism because N is normal. Therefore, \theta (H) = H and so ghg^{-1} \in H.
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