Group Representation/Examples/Quaternion Group

Example of Group Representation

Consider the group whose presentation can be expressed as:

$\Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2 = \paren {a b}^2}$

This has a matrix representation as:

\(\ds \mathbf I\) \(=\) \(\ds \begin {bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end {bmatrix}\)
\(\ds \mathbf J\) \(=\) \(\ds \begin {bmatrix} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end {bmatrix}\)
\(\ds \mathbf K\) \(=\) \(\ds \begin {bmatrix} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 \end {bmatrix}\)
\(\ds \mathbf L\) \(=\) \(\ds \begin {bmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end {bmatrix}\)

along with $-\mathbf I$, $-\mathbf J$, $-\mathbf K$ and $-\mathbf L$.


Proof

This is demonstrated in Quaternion Group/Order 4 Matrices.

$\blacksquare$


Sources

  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): representation (of a group)
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): representation (of a group)
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