# kernel

Let $\rho:G\to K$ be a group homomorphism. The preimage of the codomain identity element $e_{K}\in K$ forms a subgroup of the domain $G$, called the kernel of the homomorphism;

 $\operatorname{ker}(\rho)=\{s\in G\mid\rho(s)=e_{K}\}$

The kernel is a normal subgroup. It is the trivial subgroup if and only if $\rho$ is a monomorphism.

Title kernel Kernel 2013-03-22 11:58:24 2013-03-22 11:58:24 rmilson (146) rmilson (146) 14 rmilson (146) Definition msc 20A05 kernel of a group homomorphism GroupHomomorphism Kernel AHomomorphismIsInjectiveIffTheKernelIsTrivial