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,
if there exists a subgroup
and a normal subgroup
In the literature about sporadic groups wordings like “
“[1] can be found with the apparent meaning of “
As in the context of subgroups, in the context of subquotients the term trivial may be used for the two subquotients
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] 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.
The relation subquotient of is an order relation – which shall be denoted by
It shall be proved for groups.
φ :
be the canonical homomorphism.
φ :
are surjective for the respective pairs The preimages
φ
φ
φ
φ
As a consequence, the subquotient
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
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