Definition:Group Homomorphism

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Definition

Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups.

Let $\phi: G \to H$ be a mapping such that $\circ$ has the morphism property under $\phi$.


That is, $\forall a, b \in G$:

$\map \phi {a \circ b} = \map \phi a * \map \phi b$


Then $\phi: \struct {G, \circ} \to \struct {H, *}$ is a group homomorphism.


Also defined as

Many sources demand further that $\map \phi {e_G} = e_H$ as well, where $e_G$ and $e_H$ are the identity elements of $G$ and $H$, respectively.

However, this condition is superfluous, as shown on Group Homomorphism Preserves Identity.


Also known as

Some sources refer to a homomorphism as a representation.

Some sources use the term structure-preserving.


Examples

Mapping from Dihedral Group $D_3$ to Parity Group

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

\(\displaystyle e\) \(:\) \(\displaystyle (A) (B) (C)\) Identity mapping
\(\displaystyle p\) \(:\) \(\displaystyle (ABC)\) Rotation of $120 \degrees$ anticlockwise about center
\(\displaystyle q\) \(:\) \(\displaystyle (ACB)\) Rotation of $120 \degrees$ clockwise about center
\(\displaystyle r\) \(:\) \(\displaystyle (BC)\) Reflection in line $r$
\(\displaystyle s\) \(:\) \(\displaystyle (AC)\) Reflection in line $s$
\(\displaystyle t\) \(:\) \(\displaystyle (AB)\) Reflection in line $t$


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:

\(\displaystyle \map \ker \theta\) \(=\) \(\displaystyle \set {e, p, q}\)
\(\displaystyle \Img \theta\) \(=\) \(\displaystyle G\)


Also see


  • Results about group homomorphisms can be found here.


Linguistic Note

The word homomorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix homo- meaning similar.

Thus homomorphism means similar structure.


Sources