# Quotient Structure of Abelian Group is Abelian Group

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## 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.

$\blacksquare$

## Sources

- 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups