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Math Help - characteristic group

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    characteristic group

    If H is the unique subgroup of a given order in a group G, prove H is characteristic in G.
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    Quote Originally Posted by dori1123 View Post
    If H is the unique subgroup of a given order in a group G, prove H is characteristic in G.
    Let \theta be an automorphism of G then |\theta (H)| = |H| - why? But \phi (H) is a subgroup of H because homomorphism images of subgroups are subgroups. And by uniqueness we have \phi (H) = H. Therefore, H is invariant under all automorphism i.e. it is a characteristic subgroup.
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