Quotient Structure of Group is Group

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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 $\circ$.


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

Furthermore:

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

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.

$\blacksquare$


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