Results 1 to 2 of 2

Math Help - Factor group

  1. #1
    Member
    Joined
    Nov 2008
    Posts
    76

    Factor group

    Could someone help me on this problem?
    I want to figure out which of the following groups is isomorphic to (Z_4\times Z_{12}) / <(2,2)>. The following groups are Z_8, Z_2 \times Z_2 \times Z_2, Z_4 \times Z_2

    I know <(2,2)> has order 6, so the given group has order 48/6=8. I tried to list the elements of this factor group which has order 8, but I was unable to do so since I don't know how to pick the representatives from Z_4 \times Z_{12} so that they will give 5 distinct elements (I know <(2,2)> is the identity in this factor group). I know Z_8 is cyclic, so if I can show that the factor group has no element of order 8, then I can eliminate  Z_8. I also know every non-identity element of Z_2 \times Z_2 \times Z_2 has order 2, so if I can show that the factor group has an element of order 4, then I should be able to finish. Any help is appreciated.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Quote Originally Posted by jackie View Post
    Could someone help me on this problem?
    I want to figure out which of the following groups is isomorphic to (Z_4\times Z_{12}) / <(2,2)>. The following groups are Z_8, Z_2 \times Z_2 \times Z_2, Z_4 \times Z_2

    I know <(2,2)> has order 6, so the given group has order 48/6=8. I tried to list the elements of this factor group which has order 8, but I was unable to do so since I don't know how to pick the representatives from Z_4 \times Z_{12} so that they will give 5 distinct elements (I know <(2,2)> is the identity in this factor group). I know Z_8 is cyclic, so if I can show that the factor group has no element of order 8, then I can eliminate  Z_8. I also know every non-identity element of Z_2 \times Z_2 \times Z_2 has order 2, so if I can show that the factor group has an element of order 4, then I should be able to finish. Any help is appreciated.

    Writing \mathbb{Z}_4=\{0,1,2,3\}\!\!\!\pmod 4\,,\,\,\mathbb{Z}_{12}=\{0,1,2,\ldots, 11\}\!\!\!\pmod{12} , we get \mathbb{Z}_4\times \mathbb{Z}_{12}\geq <(2,2)>:=\{(2,2)\,,\,(0,4)\,,\,(2,6)\,,\,(0,8)\,,\  ,(2,10)\,,\,(0,0)\} .


    Now note that \forall\,b\in\mathbb{Z}_{12}\,,\,\,4b=0,4,8\!\!\!\  pmod{12} , so for any element (a,b) in the direct product we get 4(a,b) is one of (0,0)\,,\,(0,4)\,,\,(0,8) ,and all these

    elements belong to <(2,2)>\Longrightarrow the maximal order of an element in the factor group is 4...does this now give you an idea what this factor group is?

    Tonio
    Last edited by tonio; April 28th 2010 at 02:49 AM. Reason: Correction
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Factor Group
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: March 20th 2011, 04:23 PM
  2. factor group question
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 15th 2010, 06:57 PM
  3. factor group
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: March 27th 2010, 07:44 AM
  4. Factor group order
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: November 8th 2009, 02:15 PM
  5. Order of Hx in Factor Group
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: November 29th 2007, 08:34 PM

Search Tags


/mathhelpforum @mathhelpforum