symmetric subgroup of alternating groups

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October 11th 2010, 04:17 PM
Beaky

symmetric subgroup of alternating groups

Prove that $A_{n}$ contains a subgroup isomorphic to $S_{n-2}$ for all $n\geq 3$

$A_{n}$ being the alternating group of degree n and S the symmetric group.

I really have no idea how to approach this and it's had me stumped for quite a while. Any help would be much appreciated.
October 11th 2010, 07:28 PM
tonio

Quote:

Originally Posted by Beaky

Prove that $A_{n}$ contains a subgroup isomorphic to $S_{n-2}$ for all $n\geq 3$

$A_{n}$ being the alternating group of degree n and S the symmetric group.

I really have no idea how to approach this and it's had me stumped for quite a while. Any help would be much appreciated.

Say $S_{n-2}$ acts on $\{1,2,...,n-2\}$ , and $A_n\,\,on\,\,\{1,2,...,n\}$ , and let $\pi:= (n-1,\,\,n)$ be the transposition

that interchanges n-1 and n .

Define a map $f: S_{n-2}\rightarrow A_n$ by $f(\sigma):=\left\{\begin{array}{ll}\sigma&if\,\sig ma\in A_{n-2}\\\sigma\pi &if\,\sigma\in S_{n-2}-A_{n-2}\end{array}\right.$

Now show $f$ is a group monomorphism.

Tonio
October 12th 2010, 09:37 AM
Beaky

Ok thanks a lot. It seems obvious enough now, but I don't think I would have thought of that.