# Quotient Structure of Abelian Group is Abelian Group

## Theorem

Let $\RR$ be a congruence relation on an abelian group $\struct {G, \circ}$.

Then the quotient structure $\struct {G / \RR, \circ_\RR}$ is an abelian group.

## Proof

From Quotient Structure of Group is Group we have that $\struct {G / \RR, \circ_\RR}$ is a group.

Let $\eqclass x \RR, \eqclass y \RR \in S / \RR$.

 $\displaystyle \eqclass x \RR \circ_{S / \RR} \eqclass y \RR$ $=$ $\displaystyle \eqclass {x \circ y} \RR$ Definition of Operation Induced on $S / \RR$ by $\circ$ $\displaystyle$ $=$ $\displaystyle \eqclass {y \circ x} \RR$ $\circ$ is Commutative $\displaystyle$ $=$ $\displaystyle \eqclass y \RR \circ_{S / \RR} \eqclass x \RR$ Definition of Operation Induced on $S / \RR$ by $\circ$

Hence $\circ_{S / \RR}$ is commutative.

Hence $\struct {G / \RR, \circ_\RR}$ is an abelian group.

$\blacksquare$