Results 1 to 2 of 2

Math Help - How to obtain isomorphic factor groups of a group from its ismomorphic subgroups?

  1. #1
    Member
    Joined
    Sep 2009
    Posts
    161

    How to obtain isomorphic factor groups of a group from its ismomorphic subgroups?

    Dear Friends,
    If two normal subgroups H and K of a group G are ısomorphic then the factor groups G\H and G\K need not be isomorphic. My question is what are further conditions we need on H, K and probably on G that gaurantee the isomorphisms of G\H and G\K when H and K are isomorphic
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,370
    Thanks
    739

    Re: How to obtain isomorphic factor groups of a group from its ismomorphic subgroups?

    i claim the following is a sufficient (do not know about necessary) condition on H,K and G:

    there exists σ in Aut(G) with σ(H) = K.

    we would hope that this defines an isomorphism:

    σH/K from G/H to G/K, given by:

    σH/K(gH) = σ(g)K.

    first, we need to verify that σH/K is well-defined.

    suppose gH = g'H. then g'-1g is in H, so σ(g'-1g) = (σ(g'))-1σ(g) is in K, so:

    σ(g)K = σ(g')K, thus σH/K(gH) = σH/K(g'H).

    is σH/K a homomorphism?

    well, σH/K((gH)(g'H)) = σH/K(gg'H) = σ(gg')K = σ(g)σ(g')K = (σ(g)K)(σ(g')K) = σH/K(g)σH/K(g').

    is σH/K injective?

    suppose σH/K(gH) = σ(g)K = K. then σ(g) is in K, so g = σ-1(σ(g)) is in σ-1(K) = H.

    thus the the only coset of H that is in the kernel of σH/K is H.

    is σH/K surjective?

    suppose that we have ANY coset of K in G/K, say xK. let g = σ-1(x). then σH/K(gH) = σ(g)K = σ(σ-1(x))K = xK.

    ta da!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Groups, Subgroups and Normal Subgroups
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: March 28th 2012, 10:44 AM
  2. Factor groups for the quaternion group?
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: October 9th 2009, 01:29 AM
  3. Normal Subgroups and Factor Groups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 22nd 2009, 04:00 PM
  4. Group Theory: Proof on abelian and isomorphic groups.
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 20th 2009, 06:46 PM
  5. subgroups of a factor group
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: October 22nd 2008, 08:41 AM

Search Tags


/mathhelpforum @mathhelpforum