Kernel of a group homomorphism

Let $\varphi\colon G\to H$ be a group homomorphism. The kernel of $\varphi$ is the subgroup

\ker(\varphi)=\{g\in G:\varphi(g)=e_H\}.

Equivalent characterizations

Equivalently, $\ker(\varphi)$ is the preimage of $\{e_H\}$ under $\varphi$ .

Remarks

The kernel is a normal subgroup of $G$ , and $\varphi$ is injective if and only if $\ker(\varphi)=\{e_G\}$ . The first isomorphism theorem identifies $G/\ker(\varphi)$ 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 $\operatorname{sgn}\colon S_n\to\{\pm1\}$ , the kernel is $A_n$ .
For $\det\colon GL_m(\mathbb R)\to\mathbb R^\times$ , the kernel is $SL_m(\mathbb R)$ .