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Thread: Help with finding cosets of a group

  1. #1
    Newbie diehardwalnut's Avatar
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    Help with finding cosets of a group

    Let $Q$ denote the quaternian group of order 8 and let $A = \{-1,1\}$ Let $ G=\{(x,y)|x \in Q, y \in A\} $ and let $H$ denote the subgroup of $G$ generated by $(i,-1)$. What are the cosets of $H$. One coset of $H$ is just $H$ itself so $H_1 = \{e,(i,-1),(-1,1),(-i,-1)\} $. To find the other cosets do I just multiply $H$ by elements in $G$? Something like $(j,1)H$, $(j,-1)H$, $(-j,1)H$,... and so on?
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  2. #2
    MHF Contributor
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    Re: Help with finding cosets of a group

    Hey diehardwalnut.

    You need to partition the group by fixing an element of the group and finding a sub-group that meets the requirement of the coset algebra.

    So for left coset - you fix a g in G and find a H such that gH is in G and H is a subgroup of G.

    If you have a Cayley graph then fix a node in the graph and look at generating the rest of the group G with that and slowly do it for more nodes in the group.
    Thanks from diehardwalnut
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