= Quotient group
{title2=$G/N$}
{wiki}
For a <group> $G$ and a <normal subgroup> $N$, the <quotient group> $G/N$ is the <set> of <cosets> $gN$, with operation
$$
(gN)(hN)=(gh)N.
$$
Its <identity element> is the <coset> $N$, and the <inverse element> of $gN$ is $g^{-1}N$. A <quotient group> treats all elements within one <coset> as the same input while retaining a consistent operation on the classes. The <normal subgroup> condition is what makes this operation independent of which representatives are chosen. https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/ecdbd211f736b3ac92379dea05b34491_MIT18_703S13_pra_l_9.pdf[McKernan, MIT 18.703, Theorem 9.1: quotient-group construction]{ref}.
Take the additive <group> $G=\mathbb Z/6\mathbb Z$ and $N=\{0,3\}$. Since $G$ is an <abelian group>, $N$ is a <normal subgroup>. There are three elements in the <quotient group>:
$$
N=\{0,3\},\qquad 1+N=\{1,4\},\qquad 2+N=\{2,5\}.
$$
Adding two copies of $2+N$ gives $4+N=1+N$; adding $1+N$ and $2+N$ gives $3+N=N$. Hence the <quotient group> has the addition table of $\mathbb Z/3\mathbb Z$. It does not select one representative and discard the others: its elements are the entire <cosets>.
\Image[/quotient-group-mod-six.png]
{title=Identifying residues differing by three turns a six-element <cyclic group> into a three-element <quotient group>}
This is useful whenever distinctions within a <normal subgroup> should be ignored. For example, the <sign of a permutation> retains only even or odd parity of a <permutation> in $S_3$. Its <group kernel> is the <alternating group> $A_3$, and the <quotient group> $S_3/A_3$ has two elements. The <first isomorphism theorem> explains why a <group homomorphism>'s image is a <quotient group> of its domain.
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