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]

Example

There are subquotients of groups which are neither subgroup nor quotient of it. E. g. according to 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.

Order relation

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

Notation

For group ${\displaystyle G}$, subgroup ${\displaystyle G'}$ of ${\displaystyle G}$ ${\displaystyle (\Leftrightarrow :G'\leq G)}$ and normal subgroup ${\displaystyle G''}$ of ${\displaystyle G'}$ ${\displaystyle (\Leftrightarrow :G''\vartriangleleft G')}$ the quotient group ${\displaystyle H:=G'/G''}$ is a subquotient of ${\displaystyle G}$, i. e. ${\displaystyle H\preceq G}$.

1. Reflexivity: ${\displaystyle G\preceq G}$, i. e. every element is related to itself. Indeed, ${\displaystyle G}$ is isomorphic to the subquotient ${\displaystyle G/\{1\}}$ of ${\displaystyle G}$.
2. Antisymmetry: if ${\displaystyle G\preceq H}$ and ${\displaystyle H\preceq G}$ then ${\displaystyle G\cong H}$, i. e. no two distinct elements precede each other. Indeed, a comparison of the group orders of ${\displaystyle G}$ and ${\displaystyle H}$ then yields ${\displaystyle |G|=|H|}$ from which ${\displaystyle G\cong H}$.
3. Transitivity: if ${\displaystyle H'/H''\preceq H}$ and ${\displaystyle H\preceq G}$ then ${\displaystyle H'/H''\preceq G}$.

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 ${\displaystyle \varphi \colon X\to Y,\;x\mapsto x\,G''}$

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

are surjective for the respective pairs

${\displaystyle (X,Y)\;\;\;\in }$

${\displaystyle {\Bigl \{}{\bigl (}G'',\{1\}{\bigr )}{\Bigr .}}$

${\displaystyle ,}$

${\displaystyle {\bigl (}\varphi ^{-1}(H''),H''{\bigr )}}$

${\displaystyle ,}$

${\displaystyle {\bigl (}\varphi ^{-1}(H'),H'{\bigr )}}$

${\displaystyle ,}$

${\displaystyle {\Bigl .}{\bigl (}G',H{\bigr )}{\Bigr \}}.}$

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 ${\displaystyle H'/H''\cong \varphi ^{-1}\left(H'\right)/\varphi ^{-1}\left(H''\right).}$

Relation to cardinal order

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 ${\displaystyle Y}$ of ${\displaystyle X}$ is either the empty set or there is an onto function ${\displaystyle X\to Y}$. This order relation is traditionally denoted ${\displaystyle \leq ^{\ast }.}$ If additionally the axiom of choice holds, then ${\displaystyle Y}$ has a one-to-one function to ${\displaystyle X}$ and this order relation is the usual ${\displaystyle \leq }$ on corresponding cardinals.

See also

• Homological algebra
• Subcountable

References