Group homomorphism

Let $G,H$ be groups . A group homomorphism is a function $\varphi\colon G\to H$ such that for all $x,y\in G$, $\varphi(xy)=\varphi(x)\varphi(y).$ (Here $xy$ denotes the product in $G$ and $\varphi(x)\varphi(y)$ the product in $H$.)

Basic consequences of the defining identity include: $\varphi(e_G)=e_H$ and $\varphi(x^{-1})=\varphi(x)^{-1}$ for all $x\in G$. Two fundamental associated subgroups are the kernel and the image of $\varphi$. These are tied together by the first isomorphism theorem .

Examples:

• For $n\ge 1$, the reduction map $\pi\colon \mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}$ given by $\pi(k)=k\bmod n$ is a homomorphism of additive groups.
• The determinant $\det\colon GL_m(\mathbb{R})\to \mathbb{R}^\times$ is a group homomorphism under multiplication.
• If $H\le G$, the inclusion map $H\hookrightarrow G$ is a group homomorphism.