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.
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 might be called a subquotient representation; e.g., Harish-Chandra's subquotient theorem.
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 on cardinals. When one has the law of the excluded middle, then a subquotient of is either the empty set or there is an onto function. This order relation is traditionally denoted. If additionally the axiom of choice holds, then has a one-to-one function to and this order relation is the usual on corresponding cardinals.

Transitive relation

The relation »is subquotient of« is transitive.
;Proof for groups
Let be subquotient of furthermore be subquotient of and be the canonical homomorphism. Then there is
and all vertical maps with suitable are surjective for the respective pairs.
The preimages and are both subgroups of containing and it is and, because all have a preimage with. Moreover, the subgroup is a normal one.

As a consequence, the subquotient of is a subquotient of in the form.