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.

So in the algebraic structure of groups, is a subquotient of if there exists a subgroup of and a normal subgroup of so that is isomorphic to .

In the literature about sporadic groups wordings like „ is involved in [1] can be found with the apparent meaning of „ is a subquotient of “.

As in the context of subgroups, in the context of subquotients the term trivial may be used for the two subquotients and which are present in every group .[citation needed]

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]

Example

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There are subquotients of groups which are neither subgroup nor quotient of it. E. g. according to article Sporadic group, Fi22 has a double cover which is a subgroup of Fi23, so it is a subquotient of Fi23 without being a subgroup or quotient of it.

Order relation

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The relation subquotient of is an order relation – which shall be denoted by  . It shall be proved for groups.

Notation
For group  , subgroup   of     and normal subgroup   of     the quotient group   is a subquotient of  , i. e.  .
  1. Reflexivity:  , i. e. every element is related to itself. Indeed,   is isomorphic to the subquotient   of  .
  2. Antisymmetry: if   and   then  , i. e. no two distinct elements precede each other. Indeed, a comparison of the group orders of   and   then yields   from which  .
  3. Transitivity: if   and   then  .

Proof of transitivity for groups

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Let   be subquotient of  , furthermore   be subquotient of   and   be the canonical homomorphism. Then all vertical ( ) maps  

               
         
             

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 order

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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 also

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References

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  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