symmetric subgroup of alternating groups

• Oct 11th 2010, 04:17 PM
Beaky
symmetric subgroup of alternating groups
Prove that $\displaystyle A_{n}$ contains a subgroup isomorphic to $\displaystyle S_{n-2}$ for all $\displaystyle n\geq 3$

$\displaystyle A_{n}$ 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.
• Oct 11th 2010, 07:28 PM
tonio
Quote:

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

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

that interchanges n-1 and n .

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

Now show $\displaystyle f$ is a group monomorphism.

Tonio
• Oct 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.