Re: [Algebra] Group problem.. Give me some hints.please!!

From: Jim Heckman (wnzrfeurpxzna_at_lnubb.pbz.invalid)
Date: 11/02/04 

Date: Tue, 2 Nov 2004 02:01:08 GMT

On 1-Nov-2004, magidin@math.berkeley.edu (Arturo Magidin)
wrote in message <cm647k$2nnc$1@agate.berkeley.edu>:

> In article <e9263b74.0411011130.15d2d331@posting.google.com>,
> Eric <ericthms@yahoo.com> wrote:
>
> >S_n : gp permutations of order n
> >
> >If K is subgroup of S_n of index n, then K is isomorphic to S_n-1 ?
> >
> >I tried to use group action but it dosen't work well...
> >: Since S_n acts on group of left cosets S_n/K, there is an induced
> >homomorphism f: S_n -> A(S_n/K) where A(S_n/K) is permutations of
> >S_n/K , so A(S_n/K) is isomorphic to S_n. Finally, I got a
> >homomorphism f: S_n -> S_n.
>
> Which wouldn't tell you much, would it?

Huh. I would have said it tells you everything. Assuming you
accept that A_n is the only nontrivial normal subgroup of S_n
(handle n=4 separately), then f is an isomorphism to a
transitive group of degree n, so K must be isomorphic to the
point stabilizer of S_n in its transitive degree-n action,
namely S_{n-1}.

[...] 

-- 
Jim Heckman