# Quotient Structure of Abelian Group is Abelian Group

## Theorem

Let $\mathcal R$ be a congruence relation on an abelian group $\left({G, \circ}\right)$.

Then the quotient structure $\left({G / \mathcal R, \circ_\mathcal R}\right)$ is an abelian group.

## Proof

From Quotient Structure of Group is Group $\left({G / \mathcal R, \circ_\mathcal R}\right)$ is a group.

Let $\left[\!\left[{x}\right]\!\right]_\mathcal R, \left[\!\left[{y}\right]\!\right]_\mathcal R \in S / \mathcal R$.

 $\displaystyle \left[\!\left[{x}\right]\!\right]_\mathcal R \circ_{S / \mathcal R} \left[\!\left[{y}\right]\!\right]_\mathcal R$ $=$ $\displaystyle \left[\!\left[{x \circ y}\right]\!\right]_\mathcal R$ Definition of operation induced on $S / \mathcal R$ by $\circ$ $\displaystyle$ $=$ $\displaystyle \left[\!\left[{y \circ x}\right]\!\right]_\mathcal R$ $\circ$ is commutative $\displaystyle$ $=$ $\displaystyle \left[\!\left[{y}\right]\!\right]_\mathcal R \circ_{S / \mathcal R} \left[\!\left[{x}\right]\!\right]_\mathcal R$ Definition of operation induced on $S / \mathcal R$ by $\circ$

Hence $\circ_{S / \mathcal R}$ is commutative.

Hence $\left({G / \mathcal R, \circ_\mathcal R}\right)$ is an abelian group.

$\blacksquare$