# natural projection

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

 $\varphi\!:\,G\to G/H\quad\mbox{where}\quad\varphi(x)\;=\;xH\;\;\forall x\in G$

is a surjective homomorphism whose kernel is $H$.

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

 $\varphi(xy)\;=\;(xy)H\;=\;xH\!\cdot\!yH\;=\;\varphi(x)\varphi(y).$

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

 $\operatorname{ker}(\varphi)\;=\;\{x\in G\,\vdots\;\;\varphi(x)\,=\,H\}\;=\;\{x% \in G\,\vdots\;\;xH\,=\,H\}\;=\;H.$

The mapping $\varphi$ in the proposition is called natural projection or canonical homomorphism.

Title natural projection NaturalProjection 2013-03-22 19:10:16 2013-03-22 19:10:16 pahio (2872) pahio (2872) 4 pahio (2872) Definition msc 20A05 canonical homomorphism natural homomorphism QuotientGroup KernelOfAGroupHomomorphismIsANormalSubgroup