In

_{5}, the ^{''n''} include a cyclic group $\backslash mathbb/2^n\backslash mathbb$ as well as $(\backslash mathbb/2\backslash mathbb)^n$ (the _{1}, ..., ''x_{n}'' and all distinct ''y''_{1}, ..., ''y_{n}'', there is a ''g'' in ''G'' such that for . A 2-transitive action is also called ', a 3-transitive action is also called ''triply transitive'', and so on. Such actions define interesting classes of subgroups in the symmetric groups: 2-transitive groups and more generally

^{G}'': the coinvariants are a while the invariants are a . The coinvariant terminology and notation are used particularly in

^{G}'' is the zeroth

_{x}'' for all ''x'' in ''X''. If ''N'' is trivial, the action is said to be faithful (or effective).
Let ''x'' and ''y'' be two elements in ''X'', and let $g$ be a group element such that $y\; =\; g\; \backslash cdot\; x.$ Then the two stabilizer groups $G\_x$ and $G\_y$ are related by $G\_y\; =\; g\; G\_x\; g^.$ Proof: by definition, $h\; \backslash in\; G\_y$ if and only if $h\; \backslash cdot\; (g\; \backslash cdot\; x)\; =\; g\; \backslash cdot\; x.$ Applying $g^$ to both sides of this equality yields $\backslash left(g^\; hg\backslash right)\; \backslash cdot\; x\; =\; x;$ that is, $g^\; h\; g\; \backslash in\; G\_x.$ An opposite inclusion follows similarly by taking $h\; \backslash in\; G\_x$ and supposing $x\; =\; g^\; \backslash cdot\; y.$
The above says that the stabilizers of elements in the same orbit are

^{g} is the set of points fixed by ''g''. This result is mainly of use when ''G'' and ''X'' are finite, when it can be interpreted as follows: the number of orbits is equal to the average number of points fixed per group element.
Fixing a group ''G'', the set of formal differences of finite ''G''-sets forms a ring called the

_{''n''} and its subgroups act on the set by permuting its elements
* The ^{''n''}. The group operations are given by multiplying the matrices from the groups with the vectors from ''K''^{''n''}.
* The general linear group acts on Z^{''n''} by natural matrix action. The orbits of its action are classified by the greatest common divisor of coordinates of the vector in Z^{''n''}.
* The ^{n}(''K''). This is a quotient of the action of the general linear group on projective space. Particularly notable is , the symmetries of the projective line, which is sharply 3-transitive, preserving the ^{3}: for any such quaternion , the mapping is a counterclockwise rotation through an angle ''α'' about an axis given by a unit vector v; ''z'' is the same rotation; see

File:Octahedral-group-action.png, Orbit of a fundamental spherical triangle (marked in red) under action of the full octahedral group.
File:Icosahedral-group-action.png, Orbit of a fundamental spherical triangle (marked in red) under action of the full icosahedral group.

''Topology and groupoids''

Booksurge PLC, .

downloadable reprint of van Nostrand Notes in Mathematics, 1971, which deal with applications of groupoids in group theory and topology. * * * *

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, a group action on a space
Space is the boundless three-dimensional
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter
A parameter (from the Ancient Greek language, Ancient Gre ...

is a group homomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

of a given group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

into the group of transformations
Transformation may refer to:
Science and mathematics
In biology and medicine
* Metamorphosis, the biological process of changing physical form after birth or hatching
* Malignant transformation, the process of cells becoming cancerous
* Transf ...

of the space. Similarly, a group action on a mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

is a group homomorphism of a group into the automorphism group
In mathematics, the automorphism group of an object ''X'' is the group (mathematics), group consisting of automorphisms of ''X''. For example, if ''X'' is a Dimension (vector space), finite-dimensional vector space, then the automorphism group of ' ...

of the structure. It is said that the group ''acts'' on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space
Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...

and also on the figures drawn in it. In particular, it acts on the set of all triangle
A triangle is a polygon
In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The ...

s. Similarly, the group of symmetries
Symmetry (from Ancient Greek, Greek συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" ...

of a polyhedron
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

acts on the vertices, the edges, and the faces
The face is the front of an animal's head that features three of the head's Sense, sense organs, the eyes, nose, and mouth, and through which animals express many of their Emotion, emotions. The face is crucial for human Personal identity, ident ...

of the polyhedron.
A group action on a (finite-dimensional) vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

is called a representation
Representation may refer to:
Law and politics
*Representation (politics)
Political representation is the activity of making citizens "present" in public policy making processes when political actors act in the best interest of citizens. This defin ...

of the group. It allows one to identify many groups with subgroups of , the group of the invertible matrices
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and t ...

of dimension 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 grassl ...

.
The symmetric group acts on any set with elements by permuting the elements of the set. Although the group of all permutation
In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ...

s of a set depends formally on the set, the concept of group action allows one to consider a single group for studying the permutations of all sets with the same cardinality
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

.
Definition

Left group action

If is a group with identity element , and is a set, then a (''left'') ''group action'' of on is a function :$\backslash alpha\backslash colon\; G\; \backslash times\; X\; \backslash to\; X,$ that satisfies the following two axioms: : (with often shortened to or when the action being considered is clear from context): : for all and in and all in . The group is said to act on (from the left). A set together with an action of is called a (''left'') -''set''. From these two axioms, it follows that for any fixed in , the function from to itself which maps to is a bijection, with inverse bijection the corresponding map for . Therefore, one may equivalently define a group action of on as a group homomorphism from into the symmetric group of all bijections from to itself.Right group action

Likewise, a ''right group action'' of on is a function :$\backslash alpha\backslash colon\; X\; \backslash times\; G\; \backslash to\; X,$ that satisfies the analogous axioms: : (with often shortened to or when the action being considered is clear from context) : for all and in and all in . The difference between left and right actions is in the order in which a product acts on . For a left action, acts first, followed by second. For a right action, acts first, followed by second. Because of the formula , a left action can be constructed from a right action by composing with the inverse operation of the group. Also, a right action of a group on can be considered as a left action of itsopposite group
In group theory, a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathemati ...

on .
Thus, for establishing general properties of group actions, it suffices to consider only left actions. However, there are cases where this is not possible. For example, the multiplication of a group induces both a left action and a right action on the group itself—multiplication on the left and on the right, respectively.
Types of actions

The action of ''G'' on ''X'' is called: * ' if ''X'' isnon-empty#REDIRECT Empty set#REDIRECT Empty set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, chan ...

and if for each pair ''x'', ''y'' in ''X'' there exists a ''g'' in ''G'' such that . For example, the action of the symmetric group of ''X'' is transitive, the action of the general linear group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

or the special linear group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

of a vector space ''V'' on is transitive, but the action of the orthogonal group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of a Euclidean space ''E'' is not transitive on (it is transitive on the unit sphere
In mathematics, a unit sphere is simply a sphere of radius one around a given center (geometry), center. More generally, it is the Locus (mathematics), set of points of distance 1 from a fixed central point, where different norm (mathematics), norm ...

of ''E'', though).
* ' (or ') if for every two distinct ''g'', ''h'' in ''G'' there exists an ''x'' in ''X'' such that ; or equivalently, if for each in ''G'' there exists an ''x'' in ''X'' such that . In other words, in a faithful group action, different elements of ''G'' induce different permutations of ''X''. In algebraic terms, a group ''G'' acts faithfully on ''X'' if and only if the corresponding homomorphism to the symmetric group, , has a trivial kernel
Kernel may refer to:
Computing
* Kernel (operating system)
In an operating system with a Abstraction layer, layered architecture, the kernel is the lowest level, has complete control of the hardware and is always in memory. In some systems it ...

. Thus, for a faithful action, ''G'' embeds into a permutation group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

on ''X''; specifically, ''G'' is isomorphic to its image in Sym(''X''). If ''G'' does not act faithfully on ''X'', we can easily modify the group to obtain a faithful action. If we define , then ''N'' is a normal subgroup
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

of ''G''; indeed, it is the kernel of the homomorphism . The factor group
A quotient group or factor group is a math
Mathematics (from Greek: ) includes the study of such topics as quantity ( number theory), structure (algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit= ...

''G''/''N'' acts faithfully on ''X'' by setting . The original action of ''G'' on ''X'' is faithful if and only if . The smallest set on which a faithful action can be defined can vary greatly for groups of the same size. For example:
** Three groups of size 120 are the symmetric group ''S''icosahedral group
A regular icosahedron has 60 rotational (or orientation-preserving) symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation. A regular dodecahedron has the same set of symmetries, since it is th ...

, and the cyclic group $\backslash mathbb/120\backslash mathbb$. The smallest sets on which faithful actions can be defined are of size 5, 12, and 16 respectively.
** The abelian group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s of size 2direct productIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

of ''n'' copies of $\backslash mathbb/2\backslash mathbb$), but the latter acts faithfully on a set of size 2''n'', whereas the former cannot act faithfully on a set smaller than itself.
* ' (or ''semiregular'' or ''fixed-point free'') if, given ''g'', ''h'' in ''G'', the existence of an ''x'' in ''X'' with implies . Equivalently: if ''g'' is a group element and there exists an ''x'' in ''X'' with (that is, if ''g'' has at least one fixed point), then ''g'' is the identity. Note that a free action on a non-empty set is faithful.
* ' (or ' or ''sharply transitive'') if it is both transitive and free; this is equivalent to saying that for every two ''x'', ''y'' in ''X'' there exists precisely one ''g'' in ''G'' such that . In this case, ''X'' is called a principal homogeneous space:''For the term "torsor" in algebraic geometry, see torsor (algebraic geometry).''
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...

for ''G'' or a ''G''-torsor. The action of any group ''G'' on itself by left multiplication is regular, and thus faithful as well. Every group can, therefore, be embedded in the symmetric group on its own elements, Sym(''G''). This result is known as Cayley's theorem
In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group (mathematics), group ''G'' is group isomorphism, isomorphic to a subgroup of the symmetric group acting on ''G''. This can be understood as an example of ...

.
* if ''X'' has at least ''n'' elements, and for all distinct ''x''multiply transitive group
In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups Mathieu group M11, ''M''11, Mathieu group M12, ''M''12, Mathieu group M22, ''M''22, Mathieu group M23, ''M''23 and Mathieu group M24, ''M''24 intro ...

s. The action of the symmetric group on a set with ''n'' elements is always ''n''-transitive; the action of the alternating group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

is (''n'' − 2)-transitive.
* if there is exactly one such ''g''.
* ' if it is transitive and preserves no non-trivial partition of ''X''. See primitive permutation group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

for details.
* ''Locally free'' if ''G'' is a topological group
350px, The real numbers form a topological group under addition ">addition.html" ;"title="real numbers form a topological group under addition">real numbers form a topological group under addition
In mathematics, a topological group is a group ...

, and there is a neighborhood
A neighbourhood (British English
British English (BrE) is the standard dialect
A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...

''U'' of ''e'' in ''G'' such that the restriction of the action to ''U'' is free; that is, if for some ''x'' and some ''g'' in ''U'' then .
Furthermore, if ''G'' acts on a topological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

''X'', then the action is:
*'' Wandering'' if every point ''x'' in ''X'' has a neighborhood ''U'' such that $\backslash $ is finite. For example, the action of $\backslash mathbb\; Z^n$ on $\backslash mathbb\; R^n$ by translations is wandering. The action of the modular group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

on the Poincaré half-plane is also wandering.
*''Properly discontinuous'' if ''X'' is a locally compact spaceIn topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...

and for every compact subset ''K'' ⊂ ''X'' the set $\backslash $ is finite. The wandering actions given above are also properly discontinuous. On the other hand, the action of $\backslash mathbb\; Z$ on $\backslash mathbb^2\; \backslash setminus\; \backslash $ given by $n\backslash cdot\; (x,\; y)\; =\; (2^n\; x,\; 2^\; y)$ is wandering and free but not properly discontinuous.
*' if ''G'' is a topological group and the map from $G\; \backslash times\; X\; \backslash rightarrow\; X\; \backslash times\; X\; :\; (g,x)\; \backslash mapsto\; (g\; \backslash cdot\; x,x)$ is proper. If ''G'' is discrete
Discrete in science is the opposite of :wikt:continuous, continuous: something that is separate; distinct; individual.
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
*Discrete device, an electronic c ...

then properness is equivalent to proper discontinuity for ''G''-actions.
* Said to have ''discrete orbits'' if the orbit of each ''x'' in ''X'' under the action of ''G'' is discrete in ''X''.
*A ''covering space action'' if every point ''x'' in ''X'' has a neighborhood ''U'' such that $\backslash \; =\; \backslash $.
If ''X'' is a non-zero 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
* Modula ...

over a ring ''R'' and the action of ''G'' is ''R''-linear then it is said to be
* ''Irreducible'' if there is no nonzero proper invariant submodule.
Orbits and stabilizers

Consider a group ''G'' acting on a set ''X''. The ' of an element ''x'' in ''X'' is the set of elements in ''X'' to which ''x'' can be moved by the elements of ''G''. The orbit of ''x'' is denoted by $G\; \backslash cdot\; x$: $$G\; \backslash cdot\; x\; =\; \backslash .$$ The defining properties of a group guarantee that the set of orbits of (points ''x'' in) ''X'' under the action of ''G'' form a of ''X''. The associatedequivalence relation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

is defined by saying $x\; \backslash sim\; y$ if and only if
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, l ...

there exists a ''g'' in ''G'' with $g\; \backslash cdot\; x\; =\; y.$ The orbits are then the equivalence class
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

es under this relation; two elements ''x'' and ''y'' are equivalent if and only if their orbits are the same, that is, $G\; \backslash cdot\; x\; =\; G\; \backslash cdot\; y.$
The group action is transitive
Transitivity or transitive may refer to:
Grammar
* Transitivity (grammar), a property of verbs that relates to whether a verb can take direct objects
* Transitive verb, a verb which takes an object
* Transitive case, a grammatical case to mark arg ...

if and only if it has exactly one orbit, that is, if there exists ''x'' in ''X'' with $G\; \backslash cdot\; x\; =\; X.$ This is the case if and only if $G\; \backslash cdot\; x\; =\; X$ for ''x'' in ''X'' (given that ''X'' is non-empty).
The set of all orbits of ''X'' under the action of ''G'' is written as ''X''/''G'' (or, less frequently: ''G''\''X''), and is called the ' of the action. In geometric situations it may be called the ', while in algebraic situations it may be called the space of ', and written $X\_G,$ by contrast with the invariants (fixed points), denoted ''Xgroup cohomology
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

and group homology
In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study group (mathematics), groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, ...

, which use the same superscript/subscript convention.
Invariant subsets

If ''Y'' is asubset
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of ''X'', then $G\; \backslash cdot\; Y$ denotes the set $\backslash .$ The subset ''Y'' is said to be ''invariant under G'' if $G\; \backslash cdot\; Y\; =\; Y$ (which is equivalent to $G\; \backslash cdot\; Y\; \backslash subseteq\; Y$). In that case, ''G'' also operates on ''Y'' by restricting the action to ''Y''. The subset ''Y'' is called ''fixed under G'' if $g\; \backslash cdot\; y\; =\; y$ for all ''g'' in ''G'' and all ''y'' in ''Y''. Every subset that is fixed under ''G'' is also invariant under ''G'', but not conversely.
Every orbit is an invariant subset of ''X'' on which ''G'' acts transitively. Conversely, any invariant subset of ''X'' is a union of orbits. The action of ''G'' on ''X'' is ''transitive'' if and only if all elements are equivalent, meaning that there is only one orbit.
A ''G-invariant'' element of ''X'' is $x\; \backslash in\; X$ such that $g\; \backslash cdot\; x\; =\; x$ for all $g\; \backslash in\; G.$ The set of all such ''x'' is denoted $X\_G$ and called the ''G-invariants'' of ''X''. When ''X'' is a ''G''-module, ''Xcohomology
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

group of ''G'' with coefficients in ''X'', and the higher cohomology groups are the derived functorIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

s of the functor
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

of ''G''-invariants.
Fixed points and stabilizer subgroups

Given ''g'' in ''G'' and ''x'' in ''X'' with $g\; \backslash cdot\; x\; =\; x,$ it is said that "''x'' is a fixed point of ''g''" or that "''g'' fixes ''x''". For every ''x'' in ''X'', the of ''G'' with respect to ''x'' (also called the ''isotropy group'' or ''little group'') is the set of all elements in ''G'' that fix ''x'': $$G\_x\; =\; \backslash .$$ This is asubgroup
In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely ...

of ''G'', though typically not a normal one. The action of ''G'' on ''X'' is free
Free may refer to:
Concept
* Freedom, having the ability to act or change without constraint
* Emancipate, to procure political rights, as for a disenfranchised group
* Free will, control exercised by rational agents over their actions and decis ...

if and only if all stabilizers are trivial. The kernel ''N'' of the homomorphism with the symmetric group, $G\; \backslash to\; \backslash operatorname(X),$ is given by the intersection
The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points.
In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...

of the stabilizers ''Gconjugate
Conjugation or conjugate may refer to:
Linguistics
* Grammatical conjugation, the modification of a verb from its basic form
* Emotive conjugation or Russell's conjugation, the use of loaded language
Mathematics
* Complex conjugation, the change ...

to each other. Thus, to each orbit, we can associate a conjugacy class
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of a subgroup of ''G'' (that is, the set of all conjugates of the subgroup). Let $(H)$ denote the conjugacy class of ''H''. Then the orbit ''O'' has type $(H)$ if the stabilizer $G\_x$ of some/any ''x'' in ''O'' belongs to $(H)$. A maximal orbit type is often called a principal orbit type.
and Burnside's lemma

Orbits and stabilizers are closely related. For a fixed ''x'' in ''X'', consider the map $f\; :\; G\; \backslash to\; X$ given by $g\; \backslash mapsto\; g\; \backslash cdot\; x.$ By definition the image $f(G)$ of this map is the orbit $G\; \backslash cdot\; x.$ The condition for two elements to have the same image is $$f(g)=f(h)\; \backslash iff\; g\backslash cdot\; x=h\; \backslash cdot\; x\; \backslash iff\; g^h\; \backslash cdot\; x=x\; \backslash iff\; g^h\; \backslash in\; G\_x\; \backslash iff\; h\; \backslash in\; gG\_x.$$ In other words, $f(g)\; =\; f(h)$ ''if and only if'' $g$ and $h$ lie in the samecoset
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

for the stabilizer subgroup $G\_x$. Thus, the fiber
Fiber or fibre (from la, fibra, links=no) is a natural
Nature, in the broadest sense, is the natural, physical, material world or universe
The universe ( la, universus) is all of space and time and their contents, including ...

$f^(\backslash )$ of ''f'' over any ''y'' in ''G''·''x'' is contained in such a coset, and every such coset also occurs as a fiber. Therefore ''f'' defines a between the set $G/G\_x$ of cosets for the stabilizer subgroup and the orbit $G\; \backslash cdot\; x,$ which sends $gG\_x\; \backslash mapsto\; g\; \backslash cdot\; x$. This result is known as the ''orbit-stabilizer theorem''.
If ''G'' is finite then the orbit-stabilizer theorem, together with Lagrange's theorem, gives
$$,\; G\; \backslash cdot\; x,\; =;\; href="/html/ALL/s/\backslash ,:\backslash ,G\_x.html"\; ;"title="\backslash ,:\backslash ,G\_x">\backslash ,:\backslash ,G\_x$$
in other words the length of the orbit of ''x'' times the order of its stabilizer is the order of the group. In particular that implies that the orbit length is a divisor of the group order.
: Example: Let ''G'' be a group of prime order ''p'' acting on a set ''X'' with ''k'' elements. Since each orbit has either 1 or ''p'' elements, there are at least $k\; \backslash bmod\; p$ orbits of length 1 which are ''G''-invariant elements.
This result is especially useful since it can be employed for counting arguments (typically in situations where ''X'' is finite as well).
: Example: We can use the orbit-stabilizer theorem to count the automorphisms of a graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discret ...

. Consider the cubical graph
In geometry, a cube is a three-dimensional space, three-dimensional solid object bounded by six square (geometry), square faces, Facet (geometry), facets or sides, with three meeting at each vertex (geometry), vertex.
The cube is the only Reg ...

as pictured, and let ''G'' denote its automorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

group. Then ''G'' acts on the set of vertices , and this action is transitive as can be seen by composing rotations about the center of the cube. Thus, by the orbit-stabilizer theorem, $,\; G,\; =\; ,\; G\backslash cdot1,\; ,\; G\_1,\; =\; 8,\; G\_1,\; .$ Applying the theorem now to the stabilizer $G\_1,$ we can obtain $,\; G\_1,\; =\; ,\; (G\_1)\backslash cdot2,\; ,\; (G\_1)\_2,\; .$ Any element of ''G'' that fixes 1 must send 2 to either 2, 4, or 5. As an example of such automorphisms consider the rotation around the diagonal axis through 1 and 7 by $2\backslash pi/3$ which permutes 2,4,5 and 3,6,8, and fixes 1 and 7. Thus, $\backslash left,\; (G\_1)\backslash cdot2\backslash \; =\; 3.$ Applying the theorem a third time gives $,\; \backslash left(G\_1\backslash right)\_2,\; =\; ,\; \backslash left(\backslash left(G\_1\backslash right)\_2\backslash right)\backslash cdot3,\; ,\; \backslash left(\backslash left(G\_1\backslash right)\_2\backslash right)\_3,\; .$ Any element of ''G'' that fixes 1 and 2 must send 3 to either 3 or 6. Reflecting the cube at the plane through 1,2,7 and 8 is such an automorphism sending 3 to 6, thus $\backslash left,\; \backslash left(\backslash left(G\_1\backslash right)\_2\backslash right)\backslash cdot3\backslash \; =\; 2$. One also sees that $\backslash left(\backslash left(G\_1\backslash right)\_2\backslash right)\_3$ consists only of the identity automorphism, as any element of ''G'' fixing 1, 2 and 3 must also fix all other vertices, since they are determined by their adjacency to 1, 2 and 3. Combining the preceding calculations, we can now obtain $,\; G,\; =\; 8\backslash cdot3\backslash cdot2\backslash cdot1\; =\; 48.$
A result closely related to the orbit-stabilizer theorem is Burnside's lemma:
$$,\; X/G,\; =\backslash frac\backslash sum\_\; ,\; X^g,\; ,$$
where ''X''Burnside ring
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

of ''G'', where addition corresponds to disjoint union
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

, and multiplication to Cartesian product
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

.
Examples

* The ' action of any group ''G'' on any set ''X'' is defined by for all ''g'' in ''G'' and all ''x'' in ''X''; that is, every group element induces theidentity permutation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

on ''X''.
* In every group ''G'', left multiplication is an action of ''G'' on ''G'': for all ''g'', ''x'' in ''G''. This action is free and transitive (regular), and forms the basis of a rapid proof of Cayley's theorem
In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group (mathematics), group ''G'' is group isomorphism, isomorphic to a subgroup of the symmetric group acting on ''G''. This can be understood as an example of ...

- that every group is isomorphic to a subgroup of the symmetric group of permutations of the set ''G''.
* In every group ''G'' with subgroup ''H'', left multiplication is an action of ''G'' on the set of cosets ''G/H'': for all ''g'',''a'' in ''G''. In particular if H contains no nontrivial normal subgroups of ''G'' this induces an isomorphism from ''G'' to a subgroup of the permutation group of degree '' : H'.
* In every group ''G'', conjugation
Conjugation or conjugate may refer to:
Linguistics
* Grammatical conjugation, the modification of a verb from its basic form
* Emotive conjugation or Russell's conjugation, the use of loaded language
Mathematics
* Complex conjugation, the change ...

is an action of ''G'' on ''G'': . An exponential notation is commonly used for the right-action variant: ; it satisfies (.
* In every group ''G'' with subgroup ''H'', conjugation is an action of ''G'' on conjugates of ''H'': for all ''g'' in ''G'' and ''K'' conjugates of ''H''.
* The symmetric group Ssymmetry group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 ...

of a polyhedron acts on the set of vertices of that polyhedron. It also acts on the set of faces or the set of edges of the polyhedron.
* The symmetry group of any geometrical object acts on the set of points of that object.
* The automorphism group of a vector space (or graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discret ...

, or group, or ring . . .) acts on the vector space (or set of vertices of the graph, or group, or ring . . .).
* The general linear group and its subgroups, particularly its Lie subgroups (including the special linear group , orthogonal group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, special orthogonal group , and symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical Group (mathematics), groups, denoted and for positive integer ''n'' and field (mathematics), field F (usually C or R). The lat ...

) are Lie group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s that act on the vector space ''K''affine group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

acts transitively on the points of an affine space
In mathematics, an affine space is a geometric Structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping on ...

, and the subgroup V of the affine group (that is, a vector space) has transitive and free (that is, ''regular'') action on these points; indeed this can be used to give a definition of an affine space
In mathematics, an affine space is a geometric Structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping on ...

.
* The projective linear group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

and its subgroups, particularly its Lie subgroups, which are Lie groups that act on the projective space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

Pcross ratio
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, thei ...

; the Möbius group is of particular interest.
*The isometries
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

of the plane act on the set of 2D images and patterns, such as wallpaper patterns. The definition can be made more precise by specifying what is meant by image or pattern, for example, a function of position with values in a set of colors. Isometries are in fact one example of affine group (action).
*The sets acted on by a group ''G'' comprise the category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...

of ''G''-sets in which the objects are ''G''-sets and the morphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

s are ''G''-set homomorphisms: functions such that for every ''g'' in ''G''.
* The Galois group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

of a field extension
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

''L''/''K'' acts on the field L but has only a trivial action on elements of the subfield K. Subgroups of Gal(L/K) correspond to subfields of L that contain K, that is, intermediate field extensions between L and K.
* The additive group of the real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s acts on the phase space
In dynamical system theory, a phase space is a space
Space is the boundless three-dimensional
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called param ...

of "well-behaved
In mathematics, a pathological object is one which possesses deviant, irregular or counterintuitive property, in such a way that distinguishes it from what is conceived as a typical object in the same category. The opposite of pathological is ...

" systems in classical mechanics (and in more general dynamical systems
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in a Manifold, geometrical space. Examples include the mathematical models that describe the ...

) by time translation
Time translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval. Time translation symmetry is the hypothesis that the laws of physics are uncha ...

: if ''t'' is in R and ''x'' is in the phase space, then ''x'' describes a state of the system, and is defined to be the state of the system ''t'' seconds later if ''t'' is positive or −''t'' seconds ago if ''t'' is negative.
*The additive group of the real numbers acts on the set of real functions of a real variable in various ways, with (''t''⋅''f'')(''x'') equal to, for example, , , , , , or , but not .
*Given a group action of ''G'' on ''X'', we can define an induced action of ''G'' on the power set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

of ''X'', by setting for every subset ''U'' of ''X'' and every ''g'' in ''G''. This is useful, for instance, in studying the action of the large Mathieu group
In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups Mathieu group M11, ''M''11, Mathieu group M12, ''M''12, Mathieu group M22, ''M''22, Mathieu group M23, ''M''23 and Mathieu group M24, ''M''24 intro ...

on a 24-set and in studying symmetry in certain models of finite geometries.
* The quaternion
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s with norm
Norm, the Norm or NORM may refer to:
In academic disciplines
* Norm (geology), an estimate of the idealised mineral content of a rock
* Norm (philosophy)
Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...

1 (the versor
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

s), as a multiplicative group, act on Rquaternions and spatial rotation Unit
Unit may refer to:
Arts and entertainment
* UNIT
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or bea ...

. Note that this is not a faithful action because the quaternion −1 leaves all points where they were, as does the quaternion 1.
* Given left ''G''-sets $X,Y$, there is a left ''G''-set $Y^X$ whose elements are ''G''-equivariant maps $\backslash alpha:X\backslash times\; G\backslash to\; Y$, and with left ''G''-action given by $g\backslash cdot\backslash alpha=\backslash alpha\backslash circ\; (id\_X\backslash times-g)$ (where "$-g$" indicates right multiplication by $g$). This ''G''-set has the property that its fixed points correspond to equivariant maps $X\backslash to\; Y$; more generally, it is an exponential object
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

in the category of ''G''-sets.
Group actions and groupoids

The notion of group action can be put in a broader context by using the ''actiongroupoid
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

'' $G\text{'}=G\; \backslash ltimes\; X$ associated to the group action, thus allowing techniques from groupoid theory such as presentations and fibration
In topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...

s. Further the stabilizers of the action are the vertex groups, and the orbits of the action are the components, of the action groupoid. For more details, see the book ''Topology and groupoids'' referenced below.
This action groupoid comes with a morphism ''p'': ''G′'' → ''G'' which is a ''covering morphism of groupoids''. This allows a relation between such morphisms and covering maps in topology.
Morphisms and isomorphisms between ''G''-sets

If ''X'' and ''Y'' are two ''G''-sets, a ''morphism'' from ''X'' to ''Y'' is a function such that for all ''g'' in ''G'' and all ''x'' in ''X''. Morphisms of ''G''-sets are also called ''equivariant map
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s'' or ''G-maps''.
The composition of two morphisms is again a morphism. If a morphism ''f'' is bijective, then its inverse is also a morphism. In this case ''f'' is called an ''isomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

'', and the two ''G''-sets ''X'' and ''Y'' are called ''isomorphic''; for all practical purposes, isomorphic ''G''-sets are indistinguishable.
Some example isomorphisms:
* Every regular ''G'' action is isomorphic to the action of ''G'' on ''G'' given by left multiplication.
* Every free ''G'' action is isomorphic to , where ''S'' is some set and ''G'' acts on by left multiplication on the first coordinate. (''S'' can be taken to be the set of orbits ''X''/''G''.)
* Every transitive ''G'' action is isomorphic to left multiplication by ''G'' on the set of left cosets of some subgroup ''H'' of ''G''. (''H'' can be taken to be the stabilizer group of any element of the original ''G''-set.)
With this notion of morphism, the collection of all ''G''-sets forms a category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...

; this category is a Grothendieck topos
In mathematics, a topos (, ; plural topoi or , or toposes) is a category (mathematics), category that behaves like the category of Sheaf (mathematics), sheaves of Set (mathematics), sets on a topological space (or more generally: on a Site (mathem ...

(in fact, assuming a classical metalogic, this topos will even be Boolean).
Continuous group actions

One often considers ''continuous group actions'': the group ''G'' is a topological group, ''X'' is a topological space, and the map iscontinuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ga ...

with respect to the product topology
Product may refer to:
Business
* Product (business)
In marketing, a product is an object or system made available for consumer use; it is anything that can be offered to a Market (economics), market to satisfy the desire or need of a customer ...

of . The space ''X'' is also called a ''G-space'' in this case. This is indeed a generalization, since every group can be considered a topological group by using the discrete topology
In topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...

. All the concepts introduced above still work in this context, however we define morphisms between ''G''-spaces to be ''continuous'' maps compatible with the action of ''G''. The quotient ''X''/''G'' inherits the quotient topology
In topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...

from ''X'', and is called the ''quotient space'' of the action. The above statements about isomorphisms for regular, free and transitive actions are no longer valid for continuous group actions.
If ''X'' is a regular covering space of another topological space ''Y'', then the action of the deck transformation group on ''X'' is properly discontinuous as well as being free. Every free, properly discontinuous action of a group ''G'' on a path-connected
In topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...

topological space ''X'' arises in this manner: the quotient map is a regular covering map, and the deck transformation group is the given action of ''G'' on ''X''. Furthermore, if ''X'' is simply connected, the fundamental group of ''X''/''G'' will be isomorphic to ''G''.
These results have been generalized in the book ''Topology and Groupoids'' referenced below to obtain the fundamental groupoidIn algebraic topology
250px, A torus, one of the most frequently studied objects in algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebr ...

of the orbit space of a discontinuous action of a discrete group on a Hausdorff space, as, under reasonable local conditions, the orbit groupoid of the fundamental groupoid of the space. This allows calculations such as the fundamental group of the symmetric square of a space ''X'', namely the orbit space of the product of ''X'' with itself under the twist action of the cyclic group of order 2 sending to .
An action of a group ''G'' on a locally compact spaceIn topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...

''X'' is ''cocompact'' if there exists a compact subset ''A'' of ''X'' such that . For a properly discontinuous action, cocompactness is equivalent to compactness of the quotient space ''X/G''.
The action of ''G'' on ''X'' is said to be ''proper'' if the mapping that sends is a proper map
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

.
Strongly continuous group action and smooth points

A group action of a topological group ''G'' on a topological space ''X'' is said to be ''strongly continuous'' if for all ''x'' in ''X'', the map is continuous with respect to the respective topologies. Such an action induces an action on the space of continuous functions on ''X'' by defining for every ''g'' in ''G'', ''f'' a continuous function on ''X'', and ''x'' in ''X''. Note that, while every continuous group action is strongly continuous, the converse is not in general true. The subspace of ''smooth points'' for the action is the subspace of ''X'' of points ''x'' such that is smooth, that is, it is continuous and all derivatives are continuous.Variants and generalizations

We can also consider actions ofmonoid
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

s on sets, by using the same two axioms as above. This does not define bijective maps and equivalence relations however. See semigroup action
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...

.
Instead of actions on sets, we can define actions of groups and monoids on objects of an arbitrary category: start with an object ''X'' of some category, and then define an action on ''X'' as a monoid homomorphism into the monoid of endomorphisms of ''X''. If ''X'' has an underlying set, then all definitions and facts stated above can be carried over. For example, if we take the category of vector spaces, we obtain group representation
In the mathematical field of representation theory
Representation theory is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...

s in this fashion.
We can view a group ''G'' as a category with a single object in which every morphism is invertible. A (left) group action is then nothing but a (covariant) functor
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

from ''G'' to the category of sets In the mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

, and a group representation is a functor from ''G'' to the category of vector spacesIn algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In i ...

. A morphism between G-sets is then a natural transformation
In category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dir ...

between the group action functors. In analogy, an action of a groupoid
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

is a functor from the groupoid to the category of sets or to some other category.
In addition to continuous actions of topological groups on topological spaces, one also often considers smooth actions of Lie groups on smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...

s, regular actions of algebraic group
In algebraic geometry, an algebraic group (or group variety) is a Group (mathematics), group that is an algebraic variety, such that the multiplication and inversion operations are given by regular map (algebraic geometry), regular maps on the varie ...

s on algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures ...

, and actions of group schemes on scheme (mathematics), schemes. All of these are examples of group objects acting on objects of their respective category.
Gallery

See also

* Gain graph * Group with operators * Measurable group action * Monoid actionNotes

Citations

References

* * Brown, Ronald (2006)''Topology and groupoids''

Booksurge PLC, .

downloadable reprint of van Nostrand Notes in Mathematics, 1971, which deal with applications of groupoids in group theory and topology. * * * *

External links

* * {{Authority control Group theory Group actions (mathematics), Representation theory of groups Symmetry