natural projection

Proposition.  If $H$ is a normal subgroup of a group $G$, then the mapping

$$\phi :G\to G/H\mathit{\hspace{1em}}\text{where}\mathit{\hspace{1em}}\phi (x)=xH\forall x\in G$$

is a surjective homomorphism whose kernel is $H$.

Proof.  Because every coset appears as image, the mapping $\phi$ is surjective.  It is also homomorphic, since for all elements $x,y$ of $G$, one has

$$\phi (xy)=(xy)H=xH\cdot yH=\phi (x)\phi (y).$$

The identity element of the factor group $G/H$ is the coset  $eH=H$,  whence

$$\ker (\phi )=\{x\in G\mathrm{⋮}\phi (x)=H\}=\{x\in G\mathrm{⋮}xH=H\}=H.$$

The mapping $\phi$ 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