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Thread: find a subgroup H

  1. #1
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    find a subgroup H

    Find a subgroup H of sigma_8 ( permutation of {1,2,.......,8} ) such that
    H =~ Q_8 ( H is isomorphic to the quanternion group of 8)

    This is what I do:
    Draw the table of Q_8, and find the elements of the group, they are:
    1 , -1 , i , -i , k, -k , j ,-j . I dont know how to find the subgroup H.

    Can somebody please show me what i should do next in order to find the subgroup H?

    Thanks very much for your replies
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  2. #2
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    Quote Originally Posted by knguyen2005 View Post
    Find a subgroup H of sigma_8 ( permutation of {1,2,.......,8} ) such that
    H =~ Q_8 ( H is isomorphic to the quanternion group of 8)

    This is what I do:
    Draw the table of Q_8, and find the elements of the group, they are:
    1 , -1 , i , -i , k, -k , j ,-j . I dont know how to find the subgroup H.

    Can somebody please show me what i should do next in order to find the subgroup H?

    Thanks very much for your replies
    this is just a simple application of Cayley's theorem: for any $\displaystyle a \in Q_8,$ define $\displaystyle \sigma_a: Q_8 \longrightarrow Q_8$ by $\displaystyle \sigma_a(x)=ax, \ \forall x \in Q_8.$ then each $\displaystyle \sigma_a$ is a bijection. for example let's find $\displaystyle \sigma_j$:

    $\displaystyle \sigma_j=\begin{pmatrix} 1 & -1 & i & -i & j & -j & k & -k \\ j & -j & -k & k & -1 & 1 & i & -i \end{pmatrix}.$ after you found all $\displaystyle \sigma_a, \ a \in Q_8,$ rename the elements of $\displaystyle Q_8$ in $\displaystyle \sigma_a$ from $\displaystyle 1,-1,i,-i,j,-j,k,-k$ to $\displaystyle 1,2,3,4,5,6,7,8$ respectively, to get $\displaystyle \tilde{\sigma}_a$ as a

    permutation of $\displaystyle \{1,2,3,4,5,6,7,8\}.$ so for example: $\displaystyle \tilde{\sigma}_j=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 5 & 6 & 8 & 7 & 2 & 1 & 3 & 4 \end{pmatrix}=(1 \ \ 5 \ \ 2 \ \ 6)(3 \ \ 8 \ \ 4 \ \ 7).$ now the subgroup you are looking for is: $\displaystyle H=\{\tilde{\sigma}_a: \ a \in Q_8 \}.$
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