Product with Inverse on Homomorphic Image is Group Homomorphism/Examples
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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.