PlanetMath: kernel

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kernel

(Definition)

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 , called the kernel of the homomorphism;

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

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

"kernel" is owned by rmilson. [ full author list (2) | owner history (1) ]

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Other names:

kernel of a group homomorphism

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Cross-references: monomorphism, trivial subgroup, normal subgroup, domain, subgroup, identity element, codomain, preimage, group homomorphism
There are 39 references to this entry.

This is version 10 of kernel, born on 2001-11-13, modified 2004-02-25.
Object id is 812, canonical name is KernelOfAGroupHomomorphism.
Accessed 9903 times total.

Classification:

20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties)

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