Results 1 to 4 of 4

Math Help - External Direct Product isomorphism

  1. #1
    Super Member
    Joined
    Mar 2006
    Posts
    705
    Thanks
    2

    External Direct Product isomorphism

    Suppose that G_{1} \approx G_{2}, H_{1} \approx H_{2}, prove that G_{1} \oplus H_{1} \approx G_{2} \oplus H_{2}

    For this problem, I really don't have much clue, I feel I'm not too comfortable with the whole direct product thingy.

    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,957
    Thanks
    1780
    Awards
    1
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Mar 2006
    Posts
    705
    Thanks
    2
    No, I do not yet know how to solve that other problem as well. But I was hoping that by understanding how to solve this one, I might be able to solve the other one.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by tttcomrader View Post
    Suppose that G_{1} \approx G_{2}, H_{1} \approx H_{2}, prove that G_{1} \oplus H_{1} \approx G_{2} \oplus H_{2}

    For this problem, I really don't have much clue, I feel I'm not too comfortable with the whole direct product thingy.

    Thanks.
    There exists a bijective homomorphism \phi: G_1\mapsto G_2 and \psi: H_1\mapsto H_2 by definition.

    Now create G_1\times H_1 = \{(g_1,h_1)|g_1\in G_1 \mbox{ and }h_1\in H_1 and similarly G_2\times H_2.

    For x\in G_1\times H_1, i.e. x = (g_1,h_1) for some g_1\in G_1 and h_1\in H_1. Define \tau (x) = (\phi (g_1),\psi (h_1)).

    Show that \tau is a mapping from G_1\times H_1 into G_2\times H_2. Now show it is one-to-one and onto. Finally conclude by showing it is a homomorphism between these groups. It is a straightforward computation.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. External Direct Product
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: January 26th 2010, 01:42 PM
  2. External Direct Product
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: November 8th 2008, 07:47 PM
  3. External Direct Product
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: November 8th 2008, 05:29 PM
  4. Isomorphism in External Product
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 3rd 2007, 01:24 AM
  5. Isomorphism in external direct products
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 30th 2007, 04:38 AM

Search Tags


/mathhelpforum @mathhelpforum