nLab quotient group

Idea

A quotient group is a quotient object in the category Grp of groups.

Definition

For $G$ a group and $H \hookrightarrow G$ a normal subgroup, the quotient group $G/H$ is the set of cosets, equipped with a group structure induced from $G$.

Examples

• For $A \hookrightarrow B$ a morphism between abelian groups the quotient $B/A$ is equivalently the cokernel of the inclusion.

• The quotient groups of any group by itself is the trivial group: $G/G = 1$.

Related concepts

• quotient module

• quotient bimodule

• quotient ring

References

Textbook account:

• Mark A. Armstrong, chapter 15 of: Groups and Symmetry, Undergraduate Texts in Mathematics, Springer (1988) [doi:10.1007/978-1-4757-4034-9, pdf]