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 ${\displaystyle G}$" 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.

There are subquotients of groups which are neither a subgroup nor a quotient of it. For example, according to the article Sporadic group, Fi₂₂ has a double cover which is a subgroup of Fi₂₃, so it is a subquotient of Fi₂₃ without being a subgroup or quotient of it.

The relation subquotient of is an order relation, which shall be denoted by ${\displaystyle \preceq }$. It shall be proved for groups.

Let ${\displaystyle H'/H''}$ be a subquotient of ${\displaystyle H}$, let ${\displaystyle H:=G'/G''}$ be a subquotient of ${\displaystyle G}$, and let ${\displaystyle \varphi \colon G'\to H}$ be the canonical homomorphism. Then in the following diagram, all vertical (${\displaystyle \downarrow }$) maps ${\displaystyle \varphi \colon X\to Y,\;x\mapsto x\,G''}$