# find a subgroup H

• January 30th 2009, 04:49 PM
knguyen2005
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
• January 31st 2009, 04:26 PM
NonCommAlg
Quote:

Originally Posted by knguyen2005
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 $a \in Q_8,$ define $\sigma_a: Q_8 \longrightarrow Q_8$ by $\sigma_a(x)=ax, \ \forall x \in Q_8.$ then each $\sigma_a$ is a bijection. for example let's find $\sigma_j$:

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

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