Automorphism group of symmetric groups - sci.math

Message from discussion Automorphism group of symmetric groups

Arturo Magidin

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More options Nov 3 2009, 9:22 pm

Newsgroups: sci.math

From: Arturo Magidin <magi...@member.ams.org>

Date: Tue, 3 Nov 2009 13:22:10 -0800 (PST)

Local: Tues, Nov 3 2009 9:22 pm

Subject: Re: Automorphism group of symmetric groups

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On Nov 3, 3:05 pm, Al2009 <algebra_whate...@yahoo.ca> wrote:

> 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

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