Results 1 to 7 of 7
Like Tree2Thanks
  • 1 Post By Gusbob
  • 1 Post By xixi

Math Help - Isomorphism of Direct product of groups

  1. #1
    Member
    Joined
    Jul 2009
    Posts
    111
    Thanks
    1

    Lightbulb Isomorphism of Direct product of groups

    find which of the following groups is isomorphic to S3  \bigoplus Z2.

    a) Z12 b) A4 c) D6 d) Z6  \bigoplus Z2

    I eliminate option a because Z12 is cyclic whereas S3  \bigoplus Z2 is not because we know that the External direct product of G and H is cyclic if and only if the orders of G and H are relatively prime. Here it's not the case.

    Here's my question. Can I eliminate option d using the following argument?

    If S3  \bigoplus Z2 isomorphic to Z6  \bigoplus Z2 then we have S3 isomorphic to Z6, which is again a contradiction as Z6 is cyclic whereas S3 is not.

    Is my argument right?

    Also it would be great if I can get a head start with the other options too...

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Jan 2008
    Posts
    588
    Thanks
    87

    Re: Isomorphism of Direct product of groups

    An easier argument for d) is that it is abelian, whereas your original group is not. I don't think your original argument is justified.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jul 2009
    Posts
    111
    Thanks
    1

    Re: Isomorphism of Direct product of groups

    Hi Gusbob,

    I have found justification for my claim, yet I agree with you that your argument that the property of "being abelian" is a more convincing and more elegant solution. Any ideas about the other two options ... I just need to eliminate one more to arrive at the answer.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Jan 2008
    Posts
    588
    Thanks
    87

    Re: Isomorphism of Direct product of groups

    The most obvious hint is a giveaway, but I can't think of anything else at an elementary level short of writing an explicit isomorphism to the correct answer.

    S^3 is a subgroup of S^3\times Z_2. Can you realise S^3 as a subgroup of either of your two remaining options?
    Thanks from MAX09
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Jan 2010
    Posts
    59
    Thanks
    3

    Re: Isomorphism of Direct product of groups

    S_{3}\oplus Z_{2} is not isomorphic to A_{4} because the element ((123),1) has order 6 while A_{4} doesn't have any element of order 6. Actually S_{3}\oplus Z_{2} is isomorphic to the dihedral group D_{12}.
    Thanks from MAX09
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Jul 2009
    Posts
    111
    Thanks
    1

    Re: Isomorphism of Direct product of groups

    xixi, your justification was very elegant. it took a while to strike me as to why A4 should n't have an element of order 6, i realized that the order of any element of A4 is got to be the lcm of the cycles into which it can be split, which can never exceed 4 cos splitting 4 letters can only be done with at most 4 parts or lesser.

    still i guess you mean to say that the answer is D6, eh?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Jan 2010
    Posts
    59
    Thanks
    3

    Re: Isomorphism of Direct product of groups

    Yes, S_{3} \oplus Z_{2} is isomorphic to D_{12} (It is the dihedral group of order twelve) which though denoted D_{6} in an alternate convention.In other words, it is the dihedral group of degree six.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 6
    Last Post: November 24th 2012, 07:55 PM
  2. Direct Product Isomorphism
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: October 28th 2010, 12:33 AM
  3. Isomorphism of 2 groups
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: January 23rd 2010, 10:24 AM
  4. External Direct Product isomorphism
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: October 1st 2007, 12:10 PM
  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