In

^{G}'': the coinvariants are a while the invariants are a . The coinvariant terminology and notation are used particularly in group cohomology and group homology, which use the same superscript/subscript convention.

^{G}'' is the zeroth cohomology group of ''G'' with coefficients in ''X'', and the higher cohomology groups are the derived functors of the

_{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 conjugate to each other. Thus, to each orbit, we can associate a

^{''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 Burnside ring of ''G'', where addition corresponds to

_{''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 affine group acts transitively on the points of an ^{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 quaternions and spatial rotation. 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 the category of ''G''-sets.

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.

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 ...

, a group action on a space
Space is the boundless Three-dimensional space, three-dimensional extent in which Physical body, objects and events have relative position (geometry), position and direction (geometry), direction. In classical physics, physical space is often ...

is a group homomorphism
In mathematics, given two group (mathematics), groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function (mathematics), function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' ...

of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group
In 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 mod ...

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 geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

and also on the figures drawn in it. For example, it acts on the set of all triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...

s. Similarly, the group of symmetries of a polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a Three-dimensional space, three-dimensional shape with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices.
A ...

acts on the vertices, the edges, and the faces of the polyhedron.
A group action on a vector space
In 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 ...

is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of , the group of the invertible matrices
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplicati ...

of dimension over a field .
The symmetric group acts on any set with elements by permuting the elements of the set. Although the group of all permutation
In mathematics, a permutation of a Set (mathematics), set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers ...

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, the cardinality of a set (mathematics), set is a measure of the number of Element (mathematics), elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19 ...

.
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, an opposite group is a way to construct a group (mathematics), group from another group that allows one to define Group action (mathematics), right action as a special case of Group action (mathematics), ...

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.
Remarkable properties of actions

Let $G$ be a group acting on a set $X$. The action is called ' or ' if $g\; \backslash cdot\; x\; =\; x$ for all $x\; \backslash in\; X$ implies that $g\; =\; e\_G$. Equivalently, the morphism from $G$ to the group of bijections of $X$ corresponding to the action is injective. The action is called ' (or ''semiregular'' or ''fixed-point free'') if the statement that $g\; \backslash cdot\; x\; =\; x$ for some $x\; \backslash in\; X$ already implies that $g\; =\; e\_G$. In other words, no non-trivial element of $G$ fixes a point of $X$. This is a much stronger property than faithfulness. For example, the action of any group on itself by left multiplication is free. This observation impliesCayley's theorem
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 (mathemat ...

that any group can be embedded in a symmetric group (which is infinite when the group is). A finite group may act faithfully on a set of size much smaller than its cardinality (however such an action cannot be free). For instance the abelian 2-group $(\backslash mathbb\; Z/2\backslash mathbb\; Z)^n$ (of cardinality $2^n$) acts faithfully on a set of size $2n$. This is not always the case, for example the cyclic group $\backslash mathbb\; Z/2^n\backslash mathbb\; Z$ cannot act faithfully on a set of size less than $2^n$.
In general 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 $S\_5$, the icosahedral group $A\_5\; \backslash times\; \backslash mathbb\; Z/2\backslash mathbb\; Z$ and the cyclic group $\backslash mathbb\; Z\; /\; 120\backslash mathbb\; Z$. The smallest sets on which faithful actions can be defined for these groups are of size 5, 12, and 16 respectively.
Transitivity properties

The action of $G$ on $X$ is called ' if for any two points $x,\; y\; \backslash in\; X$ there exists a $g\; \backslash in\; G$ so that $g\; \backslash cdot\; x\; =\; y$. The action is ' (or ''sharply transitive'', or ') if it is both transitive and free. This means that given $x,\; y\; \backslash in\; X$ the element $g$ in the definition of transitivity is unique. If $X$ is acted upon simply transitively by a group $G$ then it is called aprincipal homogeneous space
In mathematics, a principal homogeneous space, or torsor, for a group (mathematics), group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a gr ...

for $G$ or a $G$-torsor.
For an integer $n\; \backslash ge\; 1$, the action is if $X$ has at least $n$ elements, and for any pair of $n$-tuples $(x\_1,\; \backslash ldots,\; x\_n),\; (y\_1,\; \backslash ldots,\; y\_n)\; \backslash in\; X^n$ with pairwise distinct entries (that is $x\_i\; \backslash not=x\_j$, $y\_i\; \backslash not=y\_j$ when $i\; \backslash not=\; j$) there exists a $g\; \backslash in\; G$ such that $g\; \backslash cdot\; x\_i\; =\; y\_i$ for $i=1,\backslash ldots,n$. In other words the action on the subset of $X^n$ of tuples without repeated entries is transitive. For $n=2,\; 3$ this is often called double, respectively triple, transitivity. The class of 2-transitive group A group G acts 2-transitively on a set S if it acts transitively on the set of distinct ordered pairs \. That is, assuming (without a real loss of generality) that G acts on the left of S, for each pair of pairs (x,y),(w,z)\in S\times S with x \neq ...

s (that is, subgroups of a finite symmetric group whose action is 2-transitive) and more generally 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 intr ...

s is well-studied in finite group theory.
An action is when the action on tuples without repeated entries in $X^n$ is sharply transitive.
Examples

The action of the symmetric group of $X$ is transitive, in fact $n$-transitive for any $n$ up to the cardinality of $X$. If $X$ has cardinality $n,$ the action of thealternating group
In mathematics, an alternating group is the Group (mathematics), group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted ...

is $(n-2)$-transitive but not $(n-1)$-transitive.
The action of the general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrix, invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group (mathematics), group, because the product of two in ...

of a vector space $V$ on the set $V\; \backslash setminus\; \backslash $ of non-zero vectors is transitive, but not 2-transitive (similarly for the action of the special linear group
In mathematics, the special linear group of degree ''n'' over a Field (mathematics), field ''F'' is the set of Matrix (mathematics), matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. ...

if the dimension of $v$ is at least 2). The action of the orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...

of a Euclidean space is not transitive on nonzero vectors but it is 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), n ...

.
Primitive actions

The action of $G$ on $X$ is called ''primitive'' if there is no partition of $X$ preserved by all elements of $G$ apart from the trivial partitions (the partition in a single piece and its dual, the partition into singletons).Topological properties

Assume that $X$ is atopological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

and the action of $G$ is by homeomorphism
In the mathematics, mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and Continuous function#Continuous functions between topological spaces, continuous function between topologic ...

s.
The action is ''wandering'' if every $x\; \backslash in\; X$ has a neighbourhood $U$ such that there are only finitely many $g\; \backslash in\; G$ with $g\backslash cdot\; U\; \backslash cap\; U\; \backslash not=\; \backslash emptyset$.
More generally, a point $x\; \backslash in\; X$ is called a point of discontinuity for the action of $G$ if there is an open subset $U\; \backslash ni\; x$ such that there are only finitely many $g\; \backslash in\; G$ with $g\backslash cdot\; U\; \backslash cap\; U\; \backslash not=\; \backslash emptyset$. The ''domain of discontinuity'' of the action is the set of all points of discontinuity. Equivalently it is the largest $G$-stable open subset $\backslash Omega\; \backslash subset\; X$ such that the action of $G$ on $\backslash Omega$ is wandering. In a dynamical context this is also called '' wandering set''.
The action is ''properly discontinuous'' if for every compact subset $K\; \backslash subset\; X$ there are finitely many $g\; \backslash in\; G$ such that $g\; \backslash cdot\; K\; \backslash cap\; K\; \backslash not=\; \backslash emptyset$. This is strictly stronger than wandering; for instance 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.
The action by deck transformations of the fundamental group
In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It recor ...

of a locally simply connected space on an covering space A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...

is wandering and free. Such actions can be characterized by the following property: every $x\; \backslash in\; X$ has a neighbourhood $U$ such that $g\; \backslash cdot\; U\; \backslash cap\; U\; =\; \backslash emptyset$ for every $g\; \backslash in\; G\; \backslash setminus\; \backslash $. Actions with this property are sometimes called ''freely discontinuous'', and the largest subset on which the action is freely discontinuous is then called the ''free regular set''.
An action of a group $G$ on a locally compact space In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...

$X$ is called ''cocompact'' if there exists a compact subset $A\; \backslash subset\; X$ such that $X\; =\; G\; \backslash cdot\; A$. For a properly discontinuous action, cocompactness is equivalent to compactness of the quotient space $G\; \backslash backslash\; X$.
Actions of topological groups

Now assume $G$ is atopological group
In 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 ...

and $X$ a topological space on which it acts by homeomorphisms. The action is said to be ''continuous'' if the map $G\; \backslash times\; X\; \backslash to\; X$ is continuous for the product topology.
The action is said to be ' if the map $G\; \backslash times\; X\; \backslash to\; X\; \backslash times\; X$ defined by $(g,\; x)\; \backslash mapsto\; (x,\; g\backslash cdot\; x)$ is proper. This means that given compact sets $K,\; K\text{'}$ the set of $g\; \backslash in\; G$ such that $g\; \backslash cdot\; K\; \backslash cap\; K\text{'}\; \backslash not=\; \backslash emptyset$ is compact. In particular, this is equivalent to proper discontinuity when $G$ is a discrete group
In 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 m ...

.
It is said to be ''locally free'' if there exists a neighbourhood $U$ of $e\_G$ such that $g\; \backslash cdot\; x\; \backslash not=\; x$ for all $x\; \backslash in\; X$ and $g\; \backslash in\; U\; \backslash setminus\; \backslash $.
The action is said to be ''strongly continuous'' if the orbital map $g\; \backslash mapsto\; g\; \backslash cdot\; x$ is continuous for every $x\; \backslash in\; X$. Contrary to what the name suggests, this is a weaker property than continuity of the action.
If $G$ is a Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation a ...

and $X$ a differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas (topolog ...

, then the subspace of ''smooth points'' for the action is the set of points $x\; \backslash in\; X$ such that the map $x\; \backslash mapsto\; g\; \backslash cdot\; x$ is smooth. There is a well-developed theory of Lie group actions, i.e. action which are smooth on the whole space.
Linear actions

If $g$ acts by linear transformations on a module over a commutative ring, the action is said to be irreducible if there are no proper nonzero $g$-invariant submodules. It is said to be ''semisimple'' if it decomposes as a direct sum of irreducible actions.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 partition of ''X''. The associatedequivalence relation
In 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 i ...

is defined by saying $x\; \backslash sim\; y$ if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...

there exists a ''g'' in ''G'' with $g\; \backslash cdot\; x\; =\; y.$ The orbits are then the equivalence class
In 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 ...

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 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 ''XInvariant subsets

If ''Y'' is asubset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

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, ''Xfunctor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...

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 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 of the stabilizers ''Gconjugacy class
In 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 m ...

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 same coset for the stabilizer subgroup $G\_x$. Thus, thefiber
Fiber or fibre (from la, fibra, links=no) is a #Natural fibers, natural or Fiber#Artificial fibers, artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The stronge ...

$f^(\backslash )$ 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'' induces 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. Consider the cubical graph as pictured, and let ''G'' denote its automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of map (mathematics), mapping the object to itself while preserving all of its structure. The Set (m ...

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''disjoint union
In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injective function, injection of each A_i into A, such that the image (mathematics), images of th ...

, and multiplication to Cartesian product
In mathematics, specifically set theory, the Cartesian product of two Set (mathematics), sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notatio ...

.
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 the identity permutation 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 ofCayley's theorem
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 (mathemat ...

- 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 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 symmetry group of a geometric object is the group (mathematics), group of all Transformation (geometry), transformations under which the object is invariant (mathematics), invariant, endowed with the group operation of Fu ...

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, 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, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...

, 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 la ...

) are Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation a ...

s that act on the vector space ''K''affine space
In 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 ...

, 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
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 ...

.
* The projective linear group
In 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 ...

and its subgroups, particularly its Lie subgroups, which are Lie groups that act on the projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective (graphical), perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean s ...

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, the ...

; the Möbius group is of particular interest.
*The isometries
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be Bijection, bijective. The word isometry is derived from the Ancient Greek: ἴσος ' ...

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, categories in cognitive science, information science and generally
*Category of being
*Categories (Aristotle), ''Categories'' (Aristotle)
*Category (Kant)
...

of ''G''-sets in which the objects are ''G''-sets and the morphism
In 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 mod ...

s are ''G''-set homomorphisms: functions such that for every ''g'' in ''G''.
* The Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group (mathematics), group associated with the field extension. The study of field extensions and their rel ...

of a field extension
In 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 mo ...

''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, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

s acts on the phase space
In Dynamical systems theory, dynamical system theory, a phase space is a Space (mathematics), space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. F ...

of " well-behaved" systems in classical mechanics
Classical mechanics is a Theoretical physics, physical theory describing the motion of macroscopic objects, from projectiles to parts of Machine (mechanical), machinery, and astronomical objects, such as spacecraft, planets, stars, and galax ...

(and in more general dynamical systems
In 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 i ...

) by time translation: 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 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 on a 24-set and in studying symmetry in certain models of finite geometry, finite geometries.
* The quaternions with Norm of a quaternion, norm 1 (the versors), as a multiplicative group, act on RGroup actions and groupoids

The notion of group action can be encoded by the ''action groupoid'' $G\text{'}=G\; \backslash ltimes\; X$ associated to the group action. The stabilizers of the action are the vertex groups of the groupoid and the orbits of the action are its components.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 maps'' 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'', 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 theory, category; this category is a Grothendieck topos (in fact, assuming a classical metalogic, this topos will even be Boolean).Variants and generalizations

We can also consider actions of monoids on sets, by using the same two axioms as above. This does not define bijective maps and equivalence relations however. See semigroup action. 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 representations 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, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...

from ''G'' to the category of sets, and a group representation is a functor from ''G'' to the category of vector spaces. A morphism between G-sets is then a natural transformation between the group action functors. In analogy, an action of a groupoid is a functor from the groupoid to the category of sets or to some other category.
In addition to continuous group action, continuous actions of topological groups on topological spaces, one also often considers Lie group action, smooth actions of Lie groups on manifold, smooth manifolds, regular actions of algebraic groups on algebraic variety, algebraic varieties, and group-scheme action, 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

* * * * * * * * *External links

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