KNOWPIA
WELCOME TO KNOWPIA

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 « is involved in »^{[1]} 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.^{[2]}

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".

The relation *subquotient of* is an order relation.

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.

**^**Griess, Robert L. (1982), "The Friendly Giant",*Inventiones Mathematicae*,**69**: 1−102, Bibcode:1982InMat..69....1G, doi:10.1007/BF01389186, hdl:2027.42/46608, S2CID 123597150**^**Dixmier, Jacques (1996) [1974],*Enveloping algebras*, Graduate Studies in Mathematics, vol. 11, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0560-2, MR 0498740 p. 310