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.


Examples

Square Function

Let $\struct {\R_{\ne 0}, \times}$ be the group formed from the non-zero real numbers under multiplication.

Let $\struct {\R_{>0}, \times}$ be the group formed from the (strictly) positive real numbers under multiplication.


Let $f: \R_{\ne 0} \to \R_{>0}$ be the mapping defined as:

$\forall x \in \R_{\ge 0}: \map f x = x^2$

Then $f$ is a group homomorphism.


Mapping from Dihedral Group $D_3$ to Parity Group

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

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

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


Also defined as

In their definition of a group homomorphism, some sources demand further that $\map \phi {e_G} = e_H$, 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 group homomorphism as a (group) representation.

Some sources use the term structure-preserving.


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

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