natural projection
Proposition. If is a normal subgroup of a group , then the mapping
is a surjective homomorphism whose kernel is .
Proof. Because every coset appears as image, the mapping is surjective. It is also homomorphic, since for all elements of , one has
The identity element of the factor group is the coset , whence
The mapping in the proposition is called natural projection or canonical homomorphism.
Title | natural projection |
---|---|
Canonical name | NaturalProjection |
Date of creation | 2013-03-22 19:10:16 |
Last modified on | 2013-03-22 19:10:16 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 4 |
Author | pahio (2872) |
Entry type | Definition |
Classification | msc 20A05 |
Synonym | canonical homomorphism |
Synonym | natural homomorphism |
Related topic | QuotientGroup |
Related topic | KernelOfAGroupHomomorphismIsANormalSubgroup |