In
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 branch of
mathematics
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 ...
, the
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
group with operators or Ω-group can be viewed as a
group with a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
Ω that operates on the elements of the group in a special way.
Groups with operators were extensively studied by
Emmy Noether and her school in the 1920s. She employed the concept in her original formulation of the three
Noether isomorphism theorem
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist fo ...
s.
Definition
A group with operators
can be defined as a group
together with an action of a set
on
:
:
that is
distributive relative to the group law:
:
For each
, the application
is then an
endomorphism of ''G''. From this, it results that a Ω-group can also be viewed as a group ''G'' with an
indexed family of endomorphisms of ''G''.
is called the operator domain. The associate endomorphisms are called the homotheties of ''G''.
Given two groups ''G'', ''H'' with same operator domain
, a homomorphism of groups with operators is a group homomorphism
satisfying
:
for all
and
A
subgroup ''S'' of ''G'' is called a stable subgroup,
-subgroup or
-invariant subgroup if it respects the homotheties, that is
:
for all
and
Category-theoretic remarks
In
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 ...
, a group with operators can be defined as an object of a
functor category Grp
''M'' where ''M'' is a
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids ...
(i.e. a
category with one
object
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ai ...
) and Grp denotes the
category of groups. This definition is equivalent to the previous one, provided
is a monoid (otherwise we may expand it to include the identity and all compositions).
A
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
in this category is a
natural transformation between two
functors (i.e., two groups with operators sharing same operator domain ''M''). Again we recover the definition above of a homomorphism of groups with operators (with ''f'' the
component
Circuit Component may refer to:
•Are devices that perform functions when they are connected in a circuit.
In engineering, science, and technology Generic systems
*System components, an entity with discrete structure, such as an assemb ...
of the natural transformation).
A group with operators is also a mapping
:
where
is the set of group endomorphisms of ''G''.
Examples
* Given any group ''G'', (''G'', ∅) is trivially a group with operators
* Given a
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
''M'' over a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
''R'', ''R'' acts by
scalar multiplication on the underlying
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
of ''M'', so (''M'', ''R'') is a group with operators.
* As a special case of the above, every
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''k'' is a group with operators (''V'', ''k'').
Applications
The
Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many natura ...
also holds in the context of operator groups. The requirement that a group have a
composition series is analogous to that of
compactness
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
in
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, and can sometimes be too strong a requirement. It is natural to talk about "compactness relative to a set", i.e. talk about composition series where each (
normal) subgroup is an operator-subgroup relative to the operator set ''X'', of the group in question.
See also
*
Group action
Notes
References
*
*
*{{cite book , last=Mac Lane , first=Saunders , title=Categories for the Working Mathematician , publisher=Springer-Verlag , year=1998 , isbn=0-387-98403-8
Group actions (mathematics)
Universal algebra