Group Representation

Group Representation
Faithful: Isomorphic (1 to 1)
Unfaithful: homomorphic (m to 1)

Group of Equilateral Triangle
Permute the 3 vertices.
Isomorphic to S3 (permute 1 2 3)
Faithful representation by matrices:
e = $latex
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}
$

C₃= $latex
\begin{pmatrix}
-\frac {1}{2} & \frac{\sqrt{3}}{4} \\
-\frac{\sqrt{3}}{4} & -\frac {1}{2}
\end{pmatrix}
$

C₃² = $latex
\begin{pmatrix}
-\frac {1}{2} & -\frac{\sqrt{3}}{4} \\
\frac{\sqrt{3}}{4} & -\frac {1}{2}
\end{pmatrix}
$

σ₁ = $latex
\begin{pmatrix}
-1 & 0 \\
0 & 1
\end{pmatrix}
$

σ₂ = $latex
\begin{pmatrix}
\frac {1}{2} & \frac{\sqrt{3}}{4} \\
\frac{\sqrt{3}}{4} & -\frac {1}{2}
\end{pmatrix}
$


σ₃ = $latex
\begin{pmatrix}
\frac {1}{2} & -\frac{\sqrt{3}}{4} \\
-\frac{\sqrt{3}}{4} & -\frac {1}{2}
\end{pmatrix}
$