Subquotient

In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject. Subquotients are particularly important in abelian categories, and in group theory, where they are also known as sections, though this conflicts with a different meaning in category theory.

So in the algebraic structure of groups, ${\displaystyle H}$ is a subquotient of ${\displaystyle G}$ if there exists a subgroup ${\displaystyle G'}$ of ${\displaystyle G}$ and a normal subgroup ${\displaystyle G''}$ of ${\displaystyle G'}$ so that ${\displaystyle H}$ is isomorphic to ${\displaystyle G'/G''}$.

In the literature about sporadic groups wordings like „${\displaystyle H}$ is involved in ${\displaystyle G}$“^[1] can be found with the apparent meaning of „${\displaystyle H}$ is a subquotient of ${\displaystyle G}$“.

As in the context of subgroups, in the context of subquotients the term trivial may be used for the two subquotients ${\displaystyle G}$ and ${\displaystyle \{1\}}$ which are present in every group ${\displaystyle G}$.^{[citation needed]}

A quotient of a subrepresentation of a representation (of, say, a group) might be called a subquotient representation; e. g., Harish-Chandra's subquotient theorem.^[2]