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Subquotient

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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