Subquotient

In the mathematical fields of 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 in category theory. In the literature about sporadic groups wordings like " is involved in " can be found with the apparent meaning of " is a subquotient of ." 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. Of the 26 sporadic groups, the 20 subquotients of the monster group are referred to as the "Happy Family", whereas the remaining 6 are called "pariah groups." The relation subquotient of is an order relation. Notation For group , subgroup of and normal subgroup of the quotient group is a subquotient of Let be subquotient of , furthermore be subquotient of and be the canonical homomorphism. Then all vertical () maps with suitable are surjective for the respective pairs The preimages and are both subgroups of containing and it is and , because every has a preimage with Moreover, the subgroup is normal in As a consequence, the subquotient of is a subquotient of in the form 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.