# 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.

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 ${\displaystyle G}$».

For example, of the 26 sporadic groups, the 20 subquotients of the monster group are referred to as the "Happy Family", whereas the remaining 6 as "pariah groups".

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]

In constructive set theory, where the law of excluded middle does not necessarily hold, one can consider the relation subquotient of as replacing the usual order relation(s) on cardinals. When one has the law of the excluded middle, then a subquotient ${\displaystyle X}$ of ${\displaystyle Y}$ is either the empty set or there is an onto function ${\displaystyle Y\to X}$. This order relation is traditionally denoted ${\displaystyle \leq ^{\ast }}$. If additionally the axiom of choice holds, then ${\displaystyle X}$ has a one-to-one function to ${\displaystyle Y}$ and this order relation is the usual ${\displaystyle \leq }$ on corresponding cardinals.

## Order relation

The relation subquotient of is an order relation.

Proof of transitivity for groups

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

 ${\displaystyle G}$ ${\displaystyle \geq }$ ${\displaystyle G'}$ ${\displaystyle \geq }$ ${\displaystyle \varphi ^{-1}(H')}$ ${\displaystyle \geq }$ ${\displaystyle \varphi ^{-1}(H'')}$ ${\displaystyle \vartriangleright }$ ${\displaystyle G''}$ ${\displaystyle \varphi \!:}$ ${\displaystyle {\Big \downarrow }}$ ${\displaystyle {\Big \downarrow }}$ ${\displaystyle {\Big \downarrow }}$ ${\displaystyle {\Big \downarrow }}$ ${\displaystyle H}$ ${\displaystyle \geq }$ ${\displaystyle H'}$ ${\displaystyle \vartriangleright }$ ${\displaystyle H''}$ ${\displaystyle \vartriangleright }$ ${\displaystyle \{1\}}$

with suitable ${\displaystyle g\in X}$ are surjective for the respective pairs

 ${\displaystyle (X,Y)\;\;\;\in }$ ${\displaystyle {\Bigl \{}{\bigl (}G',H{\bigr )}{\Bigr .}}$ ${\displaystyle ,}$ ${\displaystyle {\bigl (}\phi ^{-1}(H'),H'{\bigr )}}$ ${\displaystyle ,}$ ${\displaystyle {\bigl (}\phi ^{-1}(H''),H''{\bigr )}}$ ${\displaystyle ,}$ ${\displaystyle {\Bigl .}{\bigl (}G'',\{1\}{\bigr )}{\Bigr \}}.}$

The preimages ${\displaystyle \varphi ^{-1}\left(H'\right)}$ and ${\displaystyle \varphi ^{-1}\left(H''\right)}$ are both subgroups of ${\displaystyle G'}$ containing ${\displaystyle G'',}$ and it is ${\displaystyle \varphi \left(\varphi ^{-1}\left(H'\right)\right)=H'}$ and ${\displaystyle \varphi \left(\varphi ^{-1}\left(H''\right)\right)=H''}$, because every ${\displaystyle h\in H}$ has a preimage ${\displaystyle g\in G'}$ with ${\displaystyle \varphi (g)=h}$. Moreover, the subgroup ${\displaystyle \varphi ^{-1}\left(H''\right)}$ is normal in ${\displaystyle \varphi ^{-1}\left(H'\right).}$.

As a consequence, the subquotient ${\displaystyle H'/H''}$ of ${\displaystyle H}$ is a subquotient of ${\displaystyle G}$ in the form ${\displaystyle H'/H''\cong \varphi ^{-1}\left(H'\right)/\varphi ^{-1}\left(H''\right)}$.