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

Famous quotes containing the word matrix:

    ““The matrix is God?”
    “In a manner of speaking, although it would be more accurate ... to say that the matrix has a God, since this being’s omniscience and omnipotence are assumed to be limited to the matrix.”
    “If it has limits, it isn’t omnipotent.”
    “Exactly.... Cyberspace exists, insofar as it can be said to exist, by virtue of human agency.””
    —William Gibson (b. 1948)