Product with Inverse on Homomorphic Image is Group Homomorphism/Examples

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

Mapping from Dihedral Group $D_3$ to Parity Group

Let $D_3$ denote the symmetry group of the equilateral triangle:

\(\ds e\)

\(:\)

\(\ds (A) (B) (C)\)

Identity mapping

\(\ds p\)

\(:\)

\(\ds (ABC)\)

Rotation of $120 \degrees$ anticlockwise about center

\(\ds q\)

\(:\)

\(\ds (ACB)\)

Rotation of $120 \degrees$ clockwise about center

\(\ds r\)

\(:\)

\(\ds (BC)\)

Reflection in line $r$

\(\ds s\)

\(:\)

\(\ds (AC)\)

Reflection in line $s$

\(\ds t\)

\(:\)

\(\ds (AB)\)

Reflection in line $t$

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.