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Math Help - automorphisms

  1. #1
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    automorphisms

    Find a non trivial homomorphism from Aut(S_4) to S_4
    (Hint: consider the set of Sylow 3 -subgroups of S_4) , and deduce that Aut(S_4) is
    isomorphic to S_4.

    Actually we know that Inn(S_4) ~ S_4 , so S_4 < ~ Aut(S_4) .
    It seems that first question is to prove the other direction ( Aut(S_4) < S_4)
    and this can be achieved by the Cayley homomorphism.
    We have 4 syllow 3 subgroups. Let Aut(S_4) act on S the set of sylow 3 subgroups.
    so this induces f: Aut(S_4) ---> S_4 but we need to find f such that
    f is non trivial and ker(f) = {0} (i.e. f is 1-1). Any ideas ??
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  2. #2
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    I've seen this argument in another algebra forum:

    Ker(f)={g in Aut(S_4): f(g)=a_g(H)=g(H)=H}
    i.e. g fixes all Sylow 3 subgroups, and these are <(123)>, <(124)>, <(134)>, <(234)>
    and fixing <(123)> means fixing 4 and fixing <(134)> means fixing 2 and so on
    so g must fix 1,2,3,4 whence g=e. Is this argument correct ?
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