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Math Help - Find all cosets of {e,(1,2)} in S3

  1. #1
    Junior Member
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    Apr 2011
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    Find all cosets of {e,(1,2)} in S3

    Hello!
    I am hoping if someone could please explain how to do this question as I cannot get my head around it.

    Q1. Find all cosets of {e,(1,2)} in S3

    > How do I know what is in S3 if im not told in the question? will it always be s3={ e,(1,2),(1,3),(2,3),(1,2,3),(1,3,2)} (this is saying that there are 6 subgroups?)
    > Letting H={e,(1,2)} {this is saying that this is an order of 2?)
    > 6/2 = 3 therefore saying there should be 3 cosets?

    My main problem is actually solving this out.

    eH = {e,(1,2)} = He
    (1,2)H = { (1,2)(1,2)(1,2) } = {(1,2),e}= He <- How is this calculated?

    (1,3)H = { (1,3)(1,3)(1,2) } = {(1,3),(1,2,3)}= H(1,3) <- How is this calculated?


    (2,3)H = { (2,3)(2,3)(1,2) } = {(2,3),(1,3,2)}= H(2,3) <- How is this calculated?


    (1,2,3)H = { (1,2,3)(1,2,3)(1,2) } = {(1,2,3),(1,3)}= H(1,2,3) <- How is this calculated?
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  2. #2
    Junior Member
    Joined
    Apr 2011
    Posts
    29

    Re: Find all cosets of {e,(1,2)} in S3

    Hello,
    The only way I think I know how to do it, is by drawing up a s3 Cayley table and subbing in the numbers.
    When I do that, i end up with;
    12H = { 12, 13, 23 } = H12
    13H = { 13, 23, 12 } = H13
    23H = { 23, 12, 13 }
    123H = { 123, 132, e} = H(123)
    132H = { 132, e, 123}

    Therefore, 3 cosets. Could some please confirm if this is ok for me to do it like this?
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