Subquotient

Summary

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]

ExamplesEdit

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 relationEdit

The relation subquotient of is an order relation.

Proof of transitivity for groupsEdit

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  .

Relation to cardinal orderEdit

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.

See alsoEdit

ReferencesEdit

  1. ^ 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
  2. ^ 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