Results 1 to 5 of 5

Math Help - left cosets of Z2 x Z4

  1. #1
    Newbie
    Joined
    Nov 2012
    From
    Connecticut
    Posts
    5

    left cosets of Z2 x Z4

    So I am having a problem. The question is:

    Find the left coset of <(1,0)> in Z2 x Z4.

    Since there are 8 elements in Z2 x Z42 and only 2 in (1,0), there should be 4 left cosets, but I can only find one.. (1,0). What am I missing?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Ant
    Ant is offline
    Member
    Joined
    Apr 2008
    Posts
    140
    Thanks
    4

    Re: left cosets of Z2 x Z4

    Well, I'm not sure what you're missing without seeing your working out but firstly are you correctly finding the subgroup?

    <(1,0> = {(1,0), (0,0)}
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Ant
    Ant is offline
    Member
    Joined
    Apr 2008
    Posts
    140
    Thanks
    4

    Re: left cosets of Z2 x Z4

    Then, the process is:

    Take a particular element in Z_{2} x Z_{4} and add it to each of the elements in <(1,0)>. Do this for each element in Z_{2} x Z_{4}. This will give you 8 cosets, but as you say you should find there are only 4 DISJOINT cosets each with two elements.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    GJA
    GJA is offline
    Member
    Joined
    Jul 2012
    From
    USA
    Posts
    109
    Thanks
    29

    Re: left cosets of Z2 x Z4

    Hi cloeannx3,

    You're correct that there should be 4 left/right cosets - nice job! I'll work out two cosets and leave the other two for you.

    Let's start by writing out the following:

    <(1,0)> = \{(0,0), (1,0)\}

    \mathbb{Z}_{2}\times\mathbb{Z}_{4}=\{(0,0), (0,1), (0,2), (0,3), (1,0), (1,1), (1,2), (1,3)\}.

    To determine what the (left) cosets are we go down the list of elements in \mathbb{Z}_{2}\times \mathbb{Z}_{4} making cosets of <(1,0)>. The first two cosets are

    1) (0,0) + <(1,0)> = \{(0,0), (1,0)\}=<(1,0)>

    2) (0,1) + <(1,0)> = \{(0,1), (1,1)\}. Notice that this (left) coset is exactly the same as the coset (1,1) + <(1,0)>.

    Notice that the sets two cosets we have found in 1 and 2 are disjoint.

    Does this help get things on the right track? Let me know if anything is unclear. Good luck!
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Nov 2012
    From
    Connecticut
    Posts
    5

    Re: left cosets of Z2 x Z4

    That make so much more sense! Thank you so much!!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Left and Right Cosets
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 31st 2012, 06:08 PM
  2. Left and Right cosets.
    Posted in the Advanced Algebra Forum
    Replies: 12
    Last Post: February 2nd 2010, 01:35 PM
  3. Left cosets
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: January 11th 2010, 09:43 AM
  4. left and right cosets
    Posted in the Advanced Algebra Forum
    Replies: 10
    Last Post: November 4th 2009, 08:42 PM
  5. Left and Right Cosets
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 22nd 2009, 11:06 PM

Search Tags


/mathhelpforum @mathhelpforum