Thread: 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 ??

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 ?