Quotient Structure of Group is Group
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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 $1.5$