# 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 « is involved in »^{[1]} can be found with the apparent meaning of « is a subquotient of ».

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.^{[2]}

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.

## Order relation[edit]

The relation *subquotient of* is an order relation.

- Proof of transitivity for groups

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 .

## See also[edit]

## References[edit]

**^**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,**11**, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0560-2, MR 0498740 p. 310