Product with Inverse on Homomorphic Image is Group Homomorphism/Examples/Mapping from D3 to Parity Group
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Examples of Use of Product with Inverse on Homomorphic Image is Group Homomorphism
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.
Proof
\(\ds \map \ker \phi\) | \(=\) | \(\ds \set {\tuple {g_1, g_2} \in D_3 \times D_3: \map \phi {g_1, g_2} = 1}\) | Definition of Kernel of Group Homomorphism: $1$ is the identity of $G$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {\tuple {g_1, g_2} \in D_3 \times D_3: \map \theta {g_1} \map \theta {g_2}^{-1} = 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\tuple {g_1, g_2} \in D_3 \times D_3: \map \theta {g_1} \map \theta {g_2} = 1}\) | as all elements of $G$ are self-inverse | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {\tuple {g_1, g_2} \in D_3 \times D_3: \map \theta {g_1} = \map \theta {g_2} }\) | Definition of Parity Group |
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.
$\blacksquare$
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $8$: The Homomorphism Theorem: Exercise $3$