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 if there exists a subgroup and a normal subgroup is isomorphic.

In the literature about sporadic groups"".

As in the context of subgroups, in the context of subquotients the term trivial.

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.

Example

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.

Order relation

The relation subquotient of is an order relation. It shall be proved for groups.

1. Notation

2. Let be a group, let be a subgroup of , let be a normal subgroup of , and let be the quotient group . Then we say that is a subquotient of . In symbols, let and ; then . This relationship has the following properties:

3. Reflexivity, i..

4. Antisymmetry.

5. Transitivity.

Proof of transitivity for groups

be the canonical homomorphismare surjective for the respective pairsThe preimages

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 is either the empty set If additionally the axiom of choice on corresponding cardinals.

See also

• Homological algebra

• Subcountable

References