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 $\circ$.

Furthermore:

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

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