In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
fields of
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
and
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
, a subquotient is a
quotient object of a
subobject. Subquotients are particularly important in
abelian categories
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
, and in
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, where they are also known as sections, though this conflicts with
a different meaning in category theory.
In the literature about
sporadic groups
In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups.
A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. Th ...
wordings like «
is involved in
» 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
Harish-Chandra Fellow of the Royal Society, FRS (11 October 1923 – 16 October 1983) was an Indian American mathematician and physicist who did fundamental work in representation theory, especially harmonic analysis on semisimple Lie groups.
...
's subquotient theorem.
[ p. 310]
Examples
Of the 26 sporadic groups, the 20 subquotients of the
monster group
In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order
246320597611213317192329314147 ...
are referred to as the "Happy Family", whereas the remaining 6 as "
pariah group
In group theory, the term pariah was introduced by Robert Griess in to refer to the six sporadic simple groups which are not subquotients of the monster group
In the area of abstract algebra known as group theory, the monster group M (als ...
s".
Order relation
The relation ''subquotient of'' is an
order relation
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article int ...
.
Proof of transitivity for groups
Let
be subquotient of
, furthermore
be subquotient of
and
be the
canonical homomorphism
In mathematics, a canonical map, also called a natural map, is a map or morphism between objects that arises naturally from the definition or the construction of the objects. Often, it is a map which preserves the widest amount of structure. A c ...
. Then all vertical (
) maps
with suitable
are
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
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
In
constructive set theory
Constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory.
The same first-order language with "=" and "\in" of classical set theory is usually used, so this is not to be confused with a con ...
, where the
law of excluded middle
In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradic ...
does not necessarily hold, one can consider the relation ''subquotient of'' as replacing the usual
order relation
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article int ...
(s) on
cardinals
Cardinal or The Cardinal may refer to:
Animals
* Cardinal (bird) or Cardinalidae, a family of North and South American birds
**''Cardinalis'', genus of cardinal in the family Cardinalidae
**''Cardinalis cardinalis'', or northern cardinal, the ...
. When one has the law of the excluded middle, then a subquotient
of
is either the
empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
or there is an onto function
. This order relation is traditionally denoted
If additionally the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
holds, then
has a one-to-one function to
and this order relation is the usual
on corresponding cardinals.
See also
*
Homological algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
*
Subcountable
In constructive mathematics, a collection X is subcountable if there exists a partial surjection from the natural numbers onto it.
This may be expressed as
\exists (I\subseteq).\, \exists f.\, (f\colon I\twoheadrightarrow X),
where f\colon I\twohe ...
References
Category theory
Abstract algebra
{{Abstract-algebra-stub