Embed Sn in A(n+2)

[shaggymoods]

Main Question or Discussion Point

How would you embed the symmetric group on n letters in the alternating group of (n+2) letters??? I'm actually trying to write down an explicit map but can't seem to come up with one. I know An will be a subgroup of A(n+2) but I have a feeling that a map that is the identity on An and not-the-identity elsewhere won't work. Any thoughts?

 

[VKint]

Imagine a set U of n+2 symbols. A permutation p in S_n acts on the first n of these. If p is an odd permutation, define a new permutation f(p) on U by f(p) = p * t, where t is the extra transposition (n+1, n+2). Clearly, f(p) is in A_n+2. If p is an even permutation, define f(p) = p. It's easy to show that f is a homomorphism with trivial kernel.

 

[shaggymoods]

Thanks!