Homomorphism

Formal Definition

A map $\phi$ of a group $G$ into a group $G'$ is a homomorphism if the homomorphism property

(1)
\begin{align} \phi(ab)=\phi(a)\phi(b) \end{align}

holds for all $a,b\in G$

Informal Definition

A map $\phi$ of a group $\langle S, *\rangle$ into a group $\langle S', *'\rangle$ is a homomorphism if

(2)
\begin{align} \phi (a*b)=\phi (a) *'\phi (b) \end{align}

for all $a,b\in S$

Example(s)

Let $\phi: \mathbb{Z} \longrightarrow \mathbb{Z}_n$. $m \longrightarrow m\ mod\ n$ is a homomorphism.

Non-example(s)

For any groups $G$ and $G'$ ,there is always at least one homomorphism $\phi:G\rightarrow G'$,namely the trivial homomorphism defined by $\phi(g)=e'$ for all $g\in G$ , where $e'$ is the identity in $G'$.