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Math Help - Cyclic subgroup and left coset question

  1. #1
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    Cyclic subgroup and left coset question

    Hi, I have a question that I am not sure how to work out:

    Let H be the cyclic subgroup generated by g= ( 1 2 3)
    ( 1 3 2)

    Find all left cosets of S3 modulo H.

    Am i correct in that there will be two distinct left cosets, and if so how do I figure out what they are?
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  2. #2
    MHF Contributor

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    Re: Cyclic subgroup and left coset question

    First, S3 has order 6 while \begin{pmatrix}1 & 2 & 3 \\ 1 & 3 & 2\end{pmatrix} has order 2 (g just swaps 2 and 3 so doing g twice swaps back and gives the identity) so that there are 6/2= 3 left cosets. Further, those cosets partition S3 so there are, in fact three left cosets. One, the one containing the identity, is just H itself. \begin{pmatrix}1 & 2 & 3 \\ 2 & 1 & 3\end{pmatrix} is not in H and so generates another coset: taking it with the identity gives itself and taking it with \begin{pmatrix}1 & 2 & 3 \\ 1 & 3 & 2\end{pmatrix} gives \begin{pmatrix}1 & 2 & 3 \\ 3 & 1 & 2\end{pmatrix}. Can you find the third coset?
    Last edited by HallsofIvy; May 6th 2013 at 05:11 AM.
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