Quotient Structure of Group is Group

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

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

Proof
From Quotient Structure of Monoid is Monoid $\left({G / \mathcal R, \circ_\mathcal R}\right)$ is a monoid with $\left[\!\left[{e}\right]\!\right]_\mathcal R$ as its identity.

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

Consider $\left[\!\left[{-x}\right]\!\right]_\mathcal R$ where $-x$ denotes the inverse of $x$ under $\circ$.

Furthermore:

Hence $\left[\!\left[{-x}\right]\!\right]_\mathcal R$ is the inverse of $\left[\!\left[{x}\right]\!\right]_\mathcal R$.

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