Definition:Group Homomorphism

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 $t$

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.