Quotient Structure of Group is Group

Theorem

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

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

Proof

From Quotient Structure of Monoid is Monoid $\struct {G / \RR, \circ_\RR}$ is a monoid with $\eqclass e \RR$ as its identity.

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

Consider $\eqclass {-x} \RR$ where $-x$ denotes the inverse of $x$ under $\circ$.

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

\(=\)

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

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

\(\ds \)

\(=\)

\(\ds \eqclass e \RR\)

Definition of Inverse Element

Furthermore:

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

\(=\)

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

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

\(\ds \)

\(=\)

\(\ds \eqclass e \RR\)

Definition of Inverse Element

Hence $\eqclass {-x} \RR$ is the inverse of $\eqclass x \RR$.

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

$\blacksquare$

Sources

• 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups: Theorem $1.5$