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, 20 are subquotients of the monster group, and are referred to as the "Happy Family", while the other 6 are 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.

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 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

Let groups and and be group homomorphisms, then also the composition is a homomorphism.

If is a subgroup of and a subgroup of , then is a subgroup of . We have , indeed , because every has a preimage in . Thus . This means that the image, say , of a subgroup, say , of is also the image of a subgroup, namely under , of .

In other words: If is a subquotient of and is subquotient of then is subquotient of .