Subgroup of Sym(n)

1. Aug 4, 2008

dimuk

I need a proof of any subgroup of S_n which is isomorphic to S_(n-1) fixes a point in {1, 2,..., n} unless n=6.

2. Aug 4, 2008

mathwonk

the standard answer: what have you tried?

3. Aug 4, 2008

dimuk

I defined a map psi: S_n to S_(n-1) and took a subgroup H={pi \in S_n | pi(n)=n}. And proved that H is a subgroup of S_n, but I want to prove that which is isomorphic to S_(n-1) and fixes a point in {1, 2..., } unless n=6.

I thought to prove this

If X is isomorphic to S_n and Y is isomorphic to X with |X:Y|=n then Y is isomorphic to S_(n-1).

But still I don't know how to prove.

4. Aug 4, 2008

dimuk

No that second one is not correct.

5. Aug 4, 2008

dimuk

I should proof this

If n \neq 6 then any subgroup Y of S_n with |S_n:Y|=n actually fixes a point..?