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Math Help - cosets

  1. #1
    Senior Member sfspitfire23's Avatar
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    cosets

    hi all, just an interesting coset problem I'm having trouble with -

    Find the right and left cosets in G of the subgroups H and K of G.

    G=A_4; H=\{e,(1 2)(3 4),(1 3)(2 4), (1 4)(2 3)\}; K=<(1 2 3)>


    What do you guys think?
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by sfspitfire23 View Post
    hi all, just an interesting coset problem I'm having trouble with -

    Find the right and left cosets in G of the subgroups H and K of G.

    G=A_4; H=\{e,(1 2)(3 4),(1 3)(2 4), (1 4)(2 3)\}; K=<(1 2 3)>


    What do you guys think?
    Firstly, you know precisely how many right/left cosets there are for each subgroup, this is just |G|/|A| where A is your subgroup.

    Now, for left cosets we know xA=yA \Leftrightarrow x^{-1}y, and we know that cosets all have equal size (they have size |A|). Thus, you need to find however many sets of 4 elements not in H such that x^{-1}y \in H for all 4 elements.

    These will be your left cosets. For your right cosets, you apply a similar formula ( xy^{-1}).

    Remember that left and right cosets are equal if and only if your subgroup is Normal. I'm afraid I can't tell you off hand whether K is normal, although I am pretty positive it is. H is definitely normal. So, hopefully, in each case your left and right cosets will be equal.
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  3. #3
    Senior Member sfspitfire23's Avatar
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    Would e*k={e, (123),(132)}?

    I think this would be correct...but why is the (132) in there?


    thanks
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by sfspitfire23 View Post
    Would e*k={e, (123),(132)}?

    I think this would be correct...but why is the (132) in there?


    thanks
    This is correct - the coset eH is just H, as when you multiply every element in the subgroup H by the identity you get the identity. Similarly, hH=H for all h \in H. (132) is in the coset because K=\{e, (123), (132)\}=<(123)>. The <X> brackets mean "the subgroup generated by the set X". Have you come across this notation before?
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  5. #5
    Senior Member sfspitfire23's Avatar
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    ah indeed! I see now!
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