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 «formula_1 is involved in formula_2» can be found with the apparent meaning of «formula_1 is a subquotient of formula_2».
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.
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 formula_5 of formula_6 is either the empty set or there is an onto function formula_7. This order relation is traditionally denoted formula_8. If additionally the axiom of choice holds, then formula_5 has a one-to-one function to formula_6 and this order relation is the usual formula_11 on corresponding cardinals.
Order relation.
The relation "subquotient of" is an order relation.
Let formula_12 be subquotient of formula_1, furthermore formula_14 be subquotient of formula_2 and formula_16 be the canonical homomorphism. Then all vertical (formula_17) maps formula_18
with suitable formula_19 are surjective for the respective pairs
The preimages formula_20 and formula_21 are both subgroups of formula_22 containing formula_23 and it is formula_24 and formula_25, because every formula_26 has a preimage formula_27 with formula_28. Moreover, the subgroup formula_29 is normal in formula_30.
As a consequence, the subquotient formula_12 of formula_1 is a subquotient of formula_2 in the form formula_34.