Quaternion Group - Matrix Representations

Matrix Representations

The quaternion group can be represented as a subgroup of the general linear group GL2(C). A representation

is given by

1 \mapsto \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
i \mapsto \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}
j \mapsto \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}
k \mapsto \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}

Since all of the above matrices have unit determinant, this is a representation of Q in the special linear group SL2(C). The standard identities for quaternion multiplication can be verified using the usual laws of matrix multiplication in GL2(C).

There is also an important action of Q on the eight nonzero elements of the 2-dimensional vector space over the finite field F3. A representation

is given by

1 \mapsto \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
i \mapsto \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}
j \mapsto \begin{pmatrix} -1 & 1 \\ 1 & 1 \end{pmatrix}
k \mapsto \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}

where {−1,0,1} are the three elements of F3. Since all of the above matrices have unit determinant over F3, this is a representation of Q in the special linear group SL(2, 3). Indeed, the group SL(2, 3) has order 24, and Q is a normal subgroup of SL(2, 3) of index 3.

Read more about this topic:  Quaternion Group

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