< back to index

Homomorphism

Definition

Given groups $G_1, G_2$ with operations $*_1, *_2$, a map $\phi: G_1 \to G_2$ is said to be a homomorphism if the group operation is preserved, i.e. $\forall g,h \in G_1: \phi(g *_1 h) = \phi(g) *_2 \phi(h)$.

Criterion

$\forall g,h \in G_1: \phi(gh) = \phi(g)\phi(h)$. This is the same as the definition, and is usually the most efficient thing to show. The other properties such as preserving the identity fall out easily.

Properties

Basic properties

The identity is preserved, since $\phi(e_1) = \phi(e_1 e_1) = \phi(e_1) \phi(e_1)$. Inverses are also preserved.

Given homomorphisms $\phi: G_1 \to G_2$ and $\psi: G_2 \to G_3$, their composition $\phi \circ \psi: G_1 \to G_3$ is also a homomorphism.

More interesting properties

The kernel $\ker(\phi) = \phi^{-1}(e_2)$, is a normal subgroup of $G_1$. The image $\mathrm{im}\phi = \phi(G_1)$ is a subgroup of $G_2$.

The homomorphism $\phi$ is injective if and only if its kernel is trivial.

Types

A bijective homomorphism is called an isomorphism.

An isomorphism $\phi: G \to G$ is called an automorphism. The automorphisms of the group form a group $\mathrm{Aut}G$.

Related theorems

The first isomorphism theorem gives an isomorphism $G_1 / \ker\phi \cong \mathrm{im}\phi$

all these pages adapted with probably insufficient credit from my university's lecture notes