kernelKernelOfAGroupHomomorphism2001-11-13 19:40:392004-02-25 11:19:44Definition\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}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 \emph{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.