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

\(\ds \eqclass x \RR \circ_{S / \RR} \eqclass y \RR\)

\(=\)

\(\ds \eqclass {x \circ y} \RR\)

Definition of Operation Induced on $S / \RR$ by $\circ$

\(\ds \)

\(=\)

\(\ds \eqclass {y \circ x} \RR\)

$\circ$ is Commutative

\(\ds \)

\(=\)

\(\ds \eqclass y \RR \circ_{S / \RR} \eqclass x \RR\)

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