> > Hi,

> > I am trying to understand some automorphism groups of symmetric groups.

> >http://en.wikipedia.org/wiki/Symmetric_group#Automorphism_group

> > It says that

> > Aut(S_2) = C_2,
> > Aut(S_6) = S_6 \semidirect C_2
> > Aut(S_n) = S_n, for n>7.

> > I know that
> > G/Z(G) = Inn(G), Out(G) = Aut(G)/Inn(G).

> > But I can't figure out why Aut(S_6) = S_6 \semidirect C_2
> > Aut(S_n) = S_n, for n>7.

> Should be n>6.

> This is nontrivial. That Aut(S_n)=S_n for n>7 follows by looking at
> the conjugacy classes: any automorphism must send conjugacy classes to
> conjugacy classes. A simple count shows that any automorphism of S_n
> with n>6 must fix the conjugacy class of the transpositions, and then
> you can leverage that to a proof. It will also show that there is a
> possible non-inner automorphism for S_6. Constructing it is not
> obvious, however.
> --
> Arturo Magidin