Kernel of a group homomorphism

Let $\varphi\colon G\to H$ be a group homomorphism . The kernel of $\varphi$ is the subset $\ker(\varphi)=\{g\in G : \varphi(g)=e_H\}.$ Equivalently, $\ker(\varphi)$ is the preimage of $\{e_H\}$ under $\varphi$.

The kernel is always a normal subgroup of $G$. Moreover, $\varphi$ is a monomorphism if and only if $\ker(\varphi)=\{e_G\}$. The kernel controls the “collapse” of $G$ under $\varphi$; the first isomorphism theorem identifies the quotient quotient group $G/\ker(\varphi)$ with the image of $\varphi$.

Examples:

• For the reduction map $\pi\colon\mathbb{Z}\to\mathbb{Z}/n\mathbb{Z}$, one has $\ker(\pi)=n\mathbb{Z}$.
• For $\mathrm{sgn}\colon S_n\to\{\pm1\}$, the kernel is $A_n$.
• For $\det\colon GL_m(\mathbb{R})\to\mathbb{R}^\times$, the kernel is $SL_m(\mathbb{R})$.