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Math Help - Aut(H) ≈ Inn(H). Show that G ≈ H x C(H) if H is normal in G.

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    Last edited by jamiebog; May 6th 2013 at 07:51 PM.
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    Re: Aut(H) ≈ Inn(H). Show that G ≈ H x C(H) if H is normal in G.

    Hi,
    I assume you mean Inn(H) is the group of inner automorphisms of H and when you say Aut(H) \congInn(H), you really mean Aut(H)=Inn(H) -- the inner automorphisms are automorphisms. Also from your conclusion, you left out the necessary assumption that Z(H) is trivial.

    Assume H is a normal subgroup of G, Z(H) is trivial and Aut(H)=Inn(H). To show G\cong H\times C_G(H), it is sufficient to show H and CG(H) are normal, H\cap C_G(H)=<1> and G=HC_G(H). Since the centralizer is a normal subgroup and Z(H)=<1>, the only thing to prove is G=HC_G(H):

    Let x\in G. Then conjugation of elements of H by x is an automorphism of H. So there exists y\in H with x^{-1}hx=y^{-1}hy for all h\in H. That is, h=xy^{-1}hyx^{-1} for all h in H and so xy{-1}\in C_G(H). Thus x\in C_G(H)H=HC_G(H).
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