# 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 $\circ$.

 $\displaystyle \left[\!\left[{x}\right]\!\right]_\mathcal R \circ_{S / \mathcal R} \left[\!\left[{-x}\right]\!\right]_\mathcal R$ $=$ $\displaystyle \left[\!\left[{x \circ -x}\right]\!\right]_\mathcal R$ Definition of operation induced on $S / \mathcal R$ by $\circ$ $\displaystyle$ $=$ $\displaystyle \left[\!\left[{e}\right]\!\right]_\mathcal R$ Definition of Inverse Element

Furthermore:

 $\displaystyle \left[\!\left[{-x}\right]\!\right]_\mathcal R \circ_{S / \mathcal R} \left[\!\left[{x}\right]\!\right]_\mathcal R$ $=$ $\displaystyle \left[\!\left[{-x \circ x}\right]\!\right]_\mathcal R$ Definition of operation induced on $S / \mathcal R$ by $\circ$ $\displaystyle$ $=$ $\displaystyle \left[\!\left[{e}\right]\!\right]_\mathcal R$ Definition of Inverse Element

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.

$\blacksquare$