Quotient group

2020 Mathematics Subject Classification: Primary: 20-XX [MSN][ZBL]

The quotient group of a group $G$ by a normal subgroup $N$ is the group formed by the cosets (cf. Coset in a group) $Ng$, $g\in G$, of $G$; it is denoted by $G/N$ (cf. Normal subgroup). Multiplication of cosets is performed according to the formula

$$Ng_1\; Ng_2 = Ng_1g_2.$$

The unit of the quotient group is the coset $N = N\;e$, and the inverse of the coset $Ng$ is $Ng^{-1}$.

The mapping $\def\k{\kappa}\k : g\mapsto Ng$ is a group epimorphism of $G$ onto $G/N$, called the canonical epimorphism or natural epimorphism.

If $\def\phi{\varphi}\phi:G\to G'$ is an arbitrary epimorphism of $G$ onto a group $G'$, then the kernel $K$ of $\phi$ is a normal subgroup of $G$, and the quotient group $G/K$ is isomorphic to $G'$; more precisely, there is an isomorphism $\psi$ of $G/K$ onto $G'$ such that the diagram

$$\begin{matrix} G & \stackrel{\phi}\rightarrow & G'\\ & \kern-3pt\llap{\scriptstyle\kappa}\searrow & \uparrow\rlap{\scriptstyle\psi} \\ && G/K\end{matrix}$$ is commutative, where $\k$ is the natural epimorphism $G\to G/K$.

A quotient group of a group $G$ can be defined, starting from some congruence on $G$ (cf. Congruence (in algebra)), as the set of classes of congruent elements relative to multiplication of classes. All possible congruences on a group are in one-to-one correspondence with its normal subgroups, and the quotient groups by the congruences are the same as those by the normal subgroups. A quotient group is a normal quotient object in the category of groups.

References

[Co]	P.M. Cohn, "Algebra", I, Wiley (1982) pp. Sect. 9.1 MR0663370 Zbl 0481.00001