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Math Help - Cosets and subgroups

  1. #1
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    Cosets and subgroups

    Let G = U_{19}. Write down the cosets of the subgroup of G generated by [7]; by [12]; by [8]; by [2]. Verify Lagrange's theorm for each case.

    Previously G = U_m was the group (under multiplication) of units of Z/mZ

    What are the subgroups and how do I find the cosets of those subgroups?

    I really only need help getting started with one then I'm sure I'll be able to do the rest, I'm just not certain what the question is looking for.
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  2. #2
    Senior Member jakncoke's Avatar
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    for example the subgroup produced by <2> is the set of elements 2*2 mod 19,2*2*2 mod 19, 2*2*2*2 mod 19 all the way until u get your identity element, which in this group is 1. That set of elements is your subgroup produced by 2. Check that its a subgroup of U_19 in this case since your group is finite just check that <2> is closed under multiplication mod 19 (which is a subgroup test for finite groups). So  <2> = {2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1} since <2> generates the whole Group, the set of cosets is just <2> itself. For example, take  <6> = {6, 17, 7, 4, 5, 11, 9, 16, 1} now to find the cosets of the subgroup find an element in the group that isn't in <6>, take 2, now just multipily all elements of <6> with 2 under mod 19. Notice that there are 9 elements in <6> so there should be 18/9 = 2 cosets exactly such that their union produces the original group.

    so 2<6> = {12, 15, 14, 8, 10, 3, 18, 13, 2}

    so the first coset is {6, 17, 7, 4, 5, 11, 9, 16, 1}
    the second coset is 2<6> = {12, 15, 14, 8, 10, 3, 18, 13, 2}

    for the subgroup <6> in U(19)
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