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;

$$\ker (\rho )=\{s\in G\mid \rho (s)=e_K\}$$

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

Title	kernel
Canonical name	Kernel
Date of creation	2013-03-22 11:58:24
Last modified on	2013-03-22 11:58:24
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	14
Author	rmilson (146)
Entry type	Definition
Classification	msc 20A05
Synonym	kernel of a group homomorphism
Related topic	GroupHomomorphism
Related topic	Kernel
Related topic	AHomomorphismIsInjectiveIffTheKernelIsTrivial