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Math Help - AutoMorphism

  1. #1
    Junior Member mathemanyak's Avatar
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    AutoMorphism

    Let g be a finite group and let f in Aut(G) with all g in G, f(g)=g iff g=e
    Show that all g in G, there exist a in G s.t. g=a^(-1).f(a)
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  2. #2
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    Quote Originally Posted by mathemanyak View Post
    Let g be a finite group and let f in Aut(G) with all g in G, f(g)=g iff g=e
    Show that all g in G, there exist a in G s.t. g=a^(-1).f(a)
    let A=\{a^{-1}f(a): \ a \in G \}, and suppose that a^{-1}f(a)=b^{-1}f(b), for some a,b \in G. then f(ab^{-1})=ab^{-1}, and thus ab^{-1}=e, i.e.

    a=b. thus A=G. \ \ \ \square


    think about this: do we really need f to be an automorphism or being homomorphism is enough?
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