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What are cosets?

  1. Dec 13, 2004 #1
    so i'm solving problems that tell me to find the left cosets, but I dont really know what they are.

    by defn, let G be a group and H a subgp of G.. and let a be an element of G. the set ah for any h in H, denoted by aH is the left coset.

    I mean, what does that mean. so for an example problem. find the left cosets of {1, 11} in U(30). So U(30) has order 8, with elements 1 7 11 13 17 19 23 29. By formula, order of G/H equalis the number of left cosets. so 8/2 = 4. meaning we have 4 left cosets. and the book says the cosets are H 7H 13H and 19H.


    so exactly why? what are those numbers? how did they derive that?

    at first, I thought you just take each element and multiply by H, , so aH = 1H, 3H, 7H.......29H,but I guess I was way off.
     
    Last edited: Dec 13, 2004
  2. jcsd
  3. Dec 13, 2004 #2

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    I believe that is what you in fact do. But what you should find from using the multiplication table for G is that you get a second repetition of the same four cosets, i.e. you really only have four distinct cosets, not eight.
     
  4. Dec 13, 2004 #3

    ok, so if I do the multiplication table:

    1 * {1, 11} = {(1*1) (1*11)}
    7 * {1, 11} = {7*1), (7*11)}
    .
    .
    .
    .
    .
    29 *{1, 11} = {29*1) (29*11)}

    and that mod 30,

    I get
    1, 11
    7, 17
    11, 1
    13, 23
    17, 7
    19, 29
    23, 13
    29, 19

    I dotn know where the 4 distinct cosets come from
     
  5. Dec 13, 2004 #4

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    To emphasize that your eight rows of pairs are eight sets (which is what a coset is, after all), write them with brackets:

    {1, 11}
    {7, 17}
    {11, 1}
    {13, 23}
    {17, 7}
    {19, 29}
    {23, 13}
    {29, 19}

    Remember that the order that you list the elements in a set doesn't matter; it's the same set. So {1, 11} is the same set as {11, 1}, and so on. So throw out four redundant sets from your list of eight, leaving you with four distinct sets. You were 99% of the way done with the problem where you left off.
     
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