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Find the right & left cosets

  1. Mar 16, 2009 #1
    1. The problem statement, all variables and given/known data
    In each case find the right and left cosets in G of the subgroups H and K of G.
    a) G = A4; H = {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}, K = <(1 2 3)>
    b) G= Z12; H = 3Z12, K = 2Z12
    c) G = D4 = D3 = {1,a,a2
    , a3, b, ba, ba2, ba3}, |a|=4,|b|=2, and aba = b; H =<a2>, K=<b>
    d) G=any group, H is any subgroup of index 2

    2. Relevant equations
    A4 is alternating group degree 4


    3. The attempt at a solution

    a) H(1 2 3) = {(1 2 3),(2 4 3),(1 4 2),(1 3 4)} = (1 2 3)H
    H(1 3 2) = {(1 3 2),(1 4 3),(2 3 4),(1 2 4)} = (1 3 2)H
    I know this is correct, but I don't know why
    H(1 2 4) = {(1 2 4),(2 3 4),(1 4 3),(1 3 2)}...
    H(1 4 2) = {....} are not cosets of G.

    b) H + 0 = {3k | k in Z12 }
    H + 1 = {3k+1| k in Z12 }
    H + 2 = {3k+2 | k in Z12 }
    correct??

    c) Ha = {1,a,a2, a3, b, ba, ba2, ba3}
    aH = {{1,a,a2, a3, b, ab, a2b, a3b}
    K1 = {a, ab, a2b, a3b, b, ba2b, ba3b}
    1K = {{1,a,a2, a3, b, ba, ba2, ba3}
    Correct?

    d) This is just Ha and aH ?? I'm not sure about this quesiton.
     
  2. jcsd
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