Group Homomorphism/Examples/Mapping from D3 to Parity Group

Examples of Group Homomorphisms
Let $D_3$ denote the symmetry group of the equilateral triangle:


 * SymmetryGroupEqTriangle.png

Let $G$ denote the parity group, defined as:
 * $\struct {\set {1, -1}, \times}$

where $\times$ denotes conventional multiplication.

Let $\theta: D_3 \to G$ be the mapping defined as:


 * $\forall x \in D_3: \map \theta x = \begin{cases} 1 & : \text{$x$ is a rotation} \\ -1 & : \text{$x$ is a reflection} \end{cases}$

Then $\theta$ is a (group) homomorphism, where:

Proof
Let $x, y \in D_3$.

Let $P$ denote the subset of $D_3$ consisting of the rotations:
 * $P := \set {e, p, q}$

Let $R$ denote the subset of $D_3$ consisting of the reflections:
 * $R := \set {r, s, t}$

It is noted that $e$ is classified as a rotation of zero angle.

The Cayley table for $D_3$ is:

Direct reference to this Cayley table gives us:

Thus we have:

By definition, $\map \ker \phi$ is the set of all elements of $D_3$ which map to $1$.

Hence by definition:
 * $\map \ker \phi = \set {e, p, q}$

Also by definition, $\Img \phi$ is the set of all elements of $G$ which are mappped to by $\phi$.

Hence by definition:
 * $\Img \phi = \set {1, -1}$

Hence the result.