Product with Inverse on Homomorphic Image is Group Homomorphism/Examples/Mapping from D3 to Parity Group

Examples of Use of Product with Inverse on Homomorphic Image is Group Homomorphism
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 homomorphism 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}$

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


 * $\forall \tuple {g_1, g_2} \in D_3 \times D_3: \map \phi {g_1, g_2} = \map \theta {g_1} \map \theta {g_2}^{-1}$

Then the kernel $\map \ker \phi$ is the set of all pairs $\tuple {g_1, g_2}$ of elements of $D_3$ such that:
 * $g_1$ and $g_2$ are both rotations
 * $g_1$ and $g_2$ are both reflections.

Proof
Thus $\map \ker \phi$ is the set of all pairs $\tuple {g_1, g_2}$ of elements of $D_3$ such that:
 * $g_1$ and $g_2$ are both rotations
 * $g_1$ and $g_2$ are both reflections.