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». 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. Examples . 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". Order relation. The relation "subquotient of" is an order relation. Proof of transitivity for groups. Let formula_5 be subquotient of formula_1, furthermore formula_7 be subquotient of formula_2 and formula_9 be the canonical homomorphism. Then all vertical (formula_10) maps formula_11 with suitable formula_12 are surjective for the respective pairs The preimages formula_13 and formula_14 are both subgroups of formula_15 containing formula_16 and it is formula_17 and formula_18, because every formula_19 has a preimage formula_20 with  formula_21. Moreover, the subgroup formula_22 is normal in formula_23. As a consequence, the subquotient formula_5 of formula_1 is a subquotient of formula_2 in the form formula_27. 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 formula_28 of formula_29 is either the empty set or there is an onto function formula_30. This order relation is traditionally denoted formula_31 If additionally the axiom of choice holds , then formula_28 has a one-to-one function to formula_29 and this order relation is the usual formula_34 on corresponding cardinals.