Group Representation - Examples

Examples

Consider the complex number u = e2πi / 3 which has the property u3 = 1. The cyclic group C3 = {1, u, u2} has a representation ρ on C2 given by:

\rho \left( 1 \right) = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \qquad \rho \left( u \right) = \begin{bmatrix} 1 & 0 \\ 0 & u \\ \end{bmatrix} \qquad \rho \left( u^2 \right) = \begin{bmatrix} 1 & 0 \\ 0 & u^2 \\ \end{bmatrix}.

This representation is faithful because ρ is a one-to-one map.

An isomorphic representation for C3 is

\rho \left( 1 \right) = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \qquad \rho \left( u \right) = \begin{bmatrix} u & 0 \\ 0 & 1 \\ \end{bmatrix} \qquad \rho \left( u^2 \right) = \begin{bmatrix} u^2 & 0 \\ 0 & 1 \\ \end{bmatrix}.

The group C3 may also be faithfully represented on R2 by

\rho \left( 1 \right) = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \qquad \rho \left( u \right) = \begin{bmatrix} a & -b \\ b & a \\ \end{bmatrix} \qquad \rho \left( u^2 \right) = \begin{bmatrix} a & b \\ -b & a \\ \end{bmatrix}

where and .

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