Results 1 to 3 of 3

Math Help - symmetric subgroup of alternating groups

  1. #1
    Junior Member
    Joined
    Dec 2009
    Posts
    26

    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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Quote Originally Posted by Beaky View Post
    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
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Dec 2009
    Posts
    26
    Ok thanks a lot. It seems obvious enough now, but I don't think I would have thought of that.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Groups - Alternating group question
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: May 8th 2009, 11:35 PM
  2. symmetric groups 2
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 24th 2009, 09:13 AM
  3. symmetric groups 3
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 24th 2009, 09:06 AM
  4. Subgrounds of Symmetric Groups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 10th 2008, 08:11 PM
  5. Symmetric groups
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: August 6th 2008, 10:07 AM

Search Tags


/mathhelpforum @mathhelpforum