# Group action (mathematics)

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In mathematics, a **group action** is a formal way of interpreting the manner in which the elements of a group correspond to transformations of some space in a way that preserves the structure of that space. Common examples of spaces that groups act on are sets, vector spaces, and topological spaces. Actions of groups on vector spaces are called representations of the group.

For a given (finite) set, the symmetric group is an abstraction used to describe the permutations of elements of that set. The concept of group action formalizes the relationship between the group and the permutations of the set. It relates each element of the group to a particular transformation.

Subgroups of the symmetric group (including the symmetric group itself) are called **permutation groups**. A **permutation representation** of a group *G* is a representation of *G* as a group of permutations of some set, and may be described as a group representation of *G* by permutation matrices. To include non-finite cases, the concept of permutation is generalized as a bijective transformation. The bijective transformations of a set form a group, whose subgroups are **transformation groups**. An example is the group of linear transformations that act on a vector space.

**Group action** is an extension to the notion of symmetric group in which every element of the group acts as a bijective transformation on the given set, without being identified with that transformation. This allows for a more comprehensive description of transformations (such as the symmetries of a polyhedron), by allowing the same group to act on several different sets of features (such as the set of vertices, the set of edges or the set of faces of the polyhedron).

If *G* is a group and *X* is a set, then an action of *G* on *X* may be formally defined as a group homomorphism from *G* to the symmetric group of *X*. The action assigns a permutation of *X* to each element of the group in such a way that:

- the identity element of
*G*is assigned the identity transformation of*X*; - any product
*gk*of two elements of*G*is assigned the composition of the permutations assigned to*g*and*k*.

If *X* has additional structure, then is only called an action if for each , the permutation preserves the structure of *X*.

The abstraction provided by group actions is a powerful one, because it allows geometrical ideas to be applied to more abstract objects. Many objects in mathematics have natural group actions defined on them. In particular, groups can act on other groups, or even on themselves. Because of this generality, the theory of group actions contains wide-reaching theorems, such as the orbit-stabilizer theorem, which can be used to prove deep results in several fields.

## Contents

## DefinitionEdit

### Left group actionEdit

If *G* is a group and *X* is a set, then a (*left*) *group action* *φ* of *G* on *X* is a function

that satisfies the following two axioms (where we denote *φ*(*g*, *x*) as *g*⋅*x*):^{[1]}

- Identity
*e*⋅*x*=*x*for all*x*in*X*. (Here,*e*denotes the identity element of the group*G*.)- Compatibility
- (
*gh*)⋅*x*=*g*⋅(*h*⋅*x*) for all*g*,*h*in*G*and all*x*in*X*. (Here,*gh*denotes the result of applying the group operation of*G*to the elements*g*and*h*.)

The group *G* is said to act on *X* (on the left). The set *X* is called a (*left*) *G-set*.

From these two axioms, it follows that for every *g* in *G*, the function which maps *x* in *X* to *g*⋅*x* is a bijective map from *X* to *X* (its inverse being the function which maps *x* to *g*^{−1}⋅*x*). Therefore, one may alternatively define a group action of *G* on *X* as a group homomorphism from *G* into the symmetric group Sym(*X*) of all bijections from *X* to *X*.^{[2]}

### Right group actionEdit

In complete analogy, one can define a *right group action* of *G* on *X* as an operation *X* × *G* → *X* mapping (*x*, *g*) to *x*⋅*g* and satisfying the two axioms:

- Identity
*x*⋅*e*=*x*for all*x*in*X*.- Compatibility
*x*⋅(*gh*) = (*x*⋅*g*)⋅*h*for all*g*,*h*in*G*and all*x*in*X*;

The difference between left and right actions is in the order in which a product like *gh* acts on *x*. For a left action *h* acts first and is followed by *g*, while for a right action *g* acts first and is followed by *h*. Because of the formula (*gh*)^{−1} = *h*^{−1}*g*^{−1}, one can construct a left action from a right action by composing with the inverse operation of the group. Also, a right action of a group *G* on *X* is the same thing as a left action of its opposite group *G*^{op} on *X*. It is thus sufficient to only consider left actions without any loss of generality.

### Canonical mapsEdit

When there is a natural correspondence between the set of group elements and the set of space transformations, a group can be interpreted as acting on the space in a canonical way. For example, the symmetric group of a finite set consists of all bijective transformations of that set; thus, applying any element of the permutation group to an element of the set will produce another (not necessarily distinct) element of the set. More generally, symmetry groups such as the homeomorphism group of a topological space or the general linear group of a vector space, as well as their subgroups, also admit canonical actions. For other groups, an interpretation of the group in terms of an action may have to be specified, either because the group does not act canonically on any space or because the canonical action is not the action of interest. For example, we can specify an action of the two-element cyclic group on the finite set by specifying that 0 (the identity element) sends , and that 1 sends . This action is not canonical.

## Types of actionsEdit

The action of *G* on *X* is called

*Transitive*if*X*is non-empty and if for each pair*x*,*y*in*X*there exists a*g*in*G*such that*g*⋅*x*=*y*. For example, the action of the symmetric group of*X*is transitive, the action of the general linear group or the special linear group of a vector space*V*on*V*∖ {0} is transitive, but the action of the orthogonal group of a Euclidean space*E*is not transitive on*E*∖ {0} (it is transitive on the unit sphere of*E*, though).*Faithful*(or*effective*) if for every two distinct*g*,*h*in*G*there exists an*x*in*X*such that*g*⋅*x*≠*h*⋅*x*; or equivalently, if for each*g*≠*e*in*G*there exists an*x*in*X*such that*g*⋅*x*≠*x*. In other words, in a faithful group action, different elements of*G*induce different permutations of*X*.^{[a]}In algebraic terms, a group*G*acts faithfully on*X*if and only if the corresponding homomorphism to the symmetric group,*G*→ Sym(*X*), has a trivial kernel. Thus, for a faithful action,*G*embeds into a permutation group on*X*; specifically,*G*is isomorphic to its image in Sym(*X*). If*G*does not act faithfully on*X*, one can easily modify the group to obtain a faithful action. If we define*N*= {*g*in*G*:*g*⋅*x*=*x*for all*x*in*X*}, then*N*is a normal subgroup of*G*; indeed, it is the kernel of the homomorphism*G*→ Sym(*X*). The factor group*G*/*N*acts faithfully on*X*by setting (*gN*)⋅*x*=*g*⋅*x*. The original action of*G*on*X*is faithful if and only if*N*= {*e*}. 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, and the cyclic group Z_{120}. The smallest sets on which faithful actions can be defined are of size 5, 12, and 120 respectively.*Free*(or*semiregular*or*fixed point free*) if, given*g*,*h*in*G*, the existence of an*x*in*X*with*g*⋅*x*=*h*⋅*x*implies*g*=*h*. Equivalently: if*g*is a group element and there exists an*x*in*X*with*g*⋅*x*=*x*(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.*Regular*(or*simply transitive*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*g*⋅*x*=*y*. In this case,*X*is called a principal homogeneous space 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.*n-transitive*if*X*has at least*n*elements and for all pairwise distinct*x*_{1}, ...,*x*and pairwise distinct_{n}*y*_{1}, ...,*y*there is a_{n}*g*in*G*such that*g*⋅*x*=_{k}*y*for 1 ≤_{k}*k*≤*n*. A 2-transitive action is also called*doubly transitive*, 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 multiply transitive groups. The action of the symmetric group on a set with*n*elements is always*n*-transitive; the action of the alternating group is*n-2*-transitive.*Sharply n-transitive*if there is exactly one such*g*.*Primitive*if it is transitive and preserves no non-trivial partition of*X*. See primitive permutation group for details.*Locally free*if*G*is a topological group, and there is a neighbourhood*U*of*e*in*G*such that the restriction of the action to*U*is free; that is, if*g*⋅*x*=*x*for some*x*and some*g*in*U*then*g*=*e*.

Furthermore, if acts on a topological space , then the action is:

*Wandering*if every point has a neighbourhood such that is finite.^{[3]}For example, the action of on by translations is wandering. The action of the modular group on the Poincaré half-plane is also wandering.*Properly discontinuous*if is a locally compact space and for every compact subset the set is finite. The wandering actions given above are also properly discontinuous. On the other hand, the action of on by the linear map is wandering and free but not properly discontinuous.^{[4]}*Proper*if is a topological group and the map from is proper.^{[5]}If*G*is discrete then properness is equivalent to proper discontinuity for*G*-actions.- Said to have
*discrete orbits*if the orbit of each under the action of is discrete in .^{[3]} - A
*covering space action*if every point has a neighbourhood such that .^{[6]}

If *X* is a non-zero module 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 stabilizersEdit

Consider a group *G* acting on a set *X*. The *orbit* 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*⋅*x*:

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 associated equivalence relation is defined by saying *x* ∼ *y* if and only if there exists a *g* in *G* with *g*⋅*x* = *y*. The orbits are then the equivalence classes under this relation; two elements *x* and *y* are equivalent if and only if their orbits are the same, that is, *G*⋅*x* = *G*⋅*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*⋅*x* = *X*. This is the case if and only if *G*⋅*x* = *X* for *all* *x* in *X*.

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 *quotient* of the action. In geometric situations it may be called the *orbit space*, while in algebraic situations it may be called the space of *coinvariants*, and written *X _{G}*, by contrast with the invariants (fixed points), denoted

*X*: the coinvariants are a

^{G}*quotient*while the invariants are a

*subset.*The coinvariant terminology and notation are used particularly in group cohomology and group homology, which use the same superscript/subscript convention.

### Invariant subsetsEdit

If *Y* is a subset of *X*, we write *GY* for the set {*g*⋅*y* : *y* ∈ *Y* and *g* ∈ *G*}. We call the subset *Y* *invariant under G* if *G*⋅*Y* = *Y* (which is equivalent to *G*⋅*Y* ⊆ *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*⋅*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. 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* ∈ *X* such that *g*⋅*x* = *x* for all *g* ∈ *G*. The set of all such *x* is denoted *X ^{G}* and called the

*G-invariants*of

*X*. When

*X*is a

*G*-module,

*X*is the zeroth cohomology group of

^{G}*G*with coefficients in

*X*, and the higher cohomology groups are the derived functors of the functor of

*G*-invariants.

### Fixed points and stabilizer subgroupsEdit

Given *g* in *G* and *x* in *X* with *g*⋅*x* = *x*, we say *x* is a fixed point of *g* and *g* fixes *x*.

For every *x* in *X*, we define the *stabilizer subgroup* of *G* with respect to *x* (also called the *isotropy group* or *little group*^{[7]}) as the set of all elements in *G* that fix *x*:

This is a subgroup 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* → Sym(*X*), is given by the intersection of the stabilizers *G _{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*⋅*x*. Then the two stabilizer groups *G _{x}* and

*G*are related by

_{y}*G*=

_{y}*g*

*G*

_{x}*g*

^{−1}. Proof: by definition,

*h*∈

*G*if and only if

_{y}*h*⋅(

*g*⋅

*x*) =

*g*⋅

*x*. Applying

*g*

^{−1}to both sides of this equality yields (

*g*

^{−1}

*hg*)⋅

*x*=

*x*; that is,

*g*

^{−1}

*hg*∈

*G*. An opposite inclusion follows similarly by taking

_{x}*h*∈

*G*and supposing

_{x}*x*=

*g*

^{−1}⋅

*y*.

The above says that the stabilizers of elements in the same orbit are conjugate to each other. Thus, to each orbit, one can associate a conjugacy class of a subgroup of *G* (that is, the set of all conjugates of the subgroup). Let denote the conjugacy class of *H*. Then one says that the orbit *O* has type if the stabilizer of some/any *x* in *O* belongs to . A maximal orbit type is often called a principal orbit type.

### Orbit-stabilizer theorem and Burnside's lemmaEdit

This section needs expansion with: clearer explanation of what the theorem shows and how to prove it. You can help by adding to it. (June 2019) |

Orbits and stabilizers are closely related. For a fixed , consider the map given by . This map induces a bijection from the set of cosets of in to the orbit , as the translate depends only on the left coset ^{[8]}. This result is known as the *orbit-stabilizer theorem*.

If is finite then the orbit-stabilizer theorem, together with Lagrange's theorem, gives

This result is especially useful since it can be employed for counting arguments (typically in situations where is finite as well).

**Example:**One can use the orbit-stabilizer theorem to count the automorphisms of a graph. Consider the cubical graph as pictured, and let denote its automorphism group. Then 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, we have that . Applying the theorem now to the stabilizer , we obtain . Any element of that fixes must send to either , or . There are such automorphisms; consider for example the map that transposes and , transposes and , and fixes the other vertices. Thus, . Applying the theorem a third time gives . Any element of that fixes and must send to either or , and one easily finds such automorphisms. Thus, . One also sees that consists only of the identity automorphism, as any element of fixing , and must also fix and consequently all other vertices. Combining the preceding calculations, we now obtain .

A result closely related to the orbit-stabilizer theorem is Burnside's lemma:

where the set of points fixed by . This result is mainly of use when and 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 , the set of formal differences of finite -sets forms a ring called the Burnside ring of , where addition corresponds to disjoint union, and multiplication to Cartesian product.

## ExamplesEdit

- The
*trivial*action of any group*G*on any set*X*is defined by*g*⋅*x*=*x*for all*g*in*G*and all*x*in*X*; that is, every group element induces the identity permutation on*X*.^{[9]} - In every group
*G*, left multiplication is an action of*G*on*G*:*g*⋅*x*=*gx*for all*g*,*x*in*G*. This action forms the basis of a rapid proof of Cayley's theorem - 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*:*g*⋅*aH*=*gaH*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*[G : H]*. - In every group
*G*, conjugation is an action of*G*on*G*:*g*⋅*x*=*gxg*^{−1}. An exponential notation is commonly used for the right-action variant:*x*=^{g}*g*^{−1}*xg*; it satisfies (*x*^{g})^{h}=*x*^{gh}. - In every group
*G*with subgroup*H*, conjugation is an action of*G*on conjugates of*H*:*g*⋅*K*=*gKg*for all^{−1}*g*in*G*and*K*conjugates of*H*. - The symmetric group S
_{n}and its subgroups act on the set { 1, …,*n*} by permuting its elements - The symmetry group 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 GL(
*n*,*K*) and its subgroups, particularly its Lie subgroups (including the special linear group SL(*n*,*K*), orthogonal group O(*n*,*K*), special orthogonal group SO(*n*,*K*), and symplectic group Sp(*n*,*K*)) are Lie groups that act on the vector space*K*^{n}. The group operations are given by multiplying the matrices from the groups with the vectors from*K*^{n}. - The general linear group GL(
*n*,**Z**) 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 affine space, and the subgroup V of the affine group (that is, a vector space) transitive and free (that is,
*regular*) action on these points;^{[10]}indeed this can be used to give a definition of an affine space. - The projective linear group PGL(
*n*+ 1,*K*) and its subgroups, particularly its Lie subgroups, which are Lie groups that act on the projective space**P**^{n}(*K*). This is a quotient of the action of the general linear group on projective space. Particularly notable is PGL(2,*K*), the symmetries of the projective line, which is sharply 3-transitive, preserving the cross ratio; the Möbius group PGL(2,**C**) is of particular interest. - The isometries 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).
^{[dubious – discuss]} - The sets acted on by a group G comprise the category of G-sets in which the objects are G-sets and the morphisms are G-set homomorphisms: functions
*f*:*X*→*Y*such that*g*⋅(*f*(*x*)) =*f*(*g*⋅*x*) for every*g*in*G*. - The Galois group of a field extension
*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 numbers (
**R**, +) acts on the phase space of "well-behaved" systems in classical mechanics (and in more general dynamical systems) by time translation: if*t*is in**R**and*x*is in the phase space, then*x*describes a state of the system, and*t*+*x*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 (
**R**, +) acts on the set of real functions of a real variable in various ways, with (*t*⋅*f*)(*x*) equal to, for example,*f*(*x*+*t*),*f*(*x*) +*t*,*f*(*xe*),^{t}*f*(*x*)*e*,^{t}*f*(*x*+*t*)*e*, or^{t}*f*(*xe*) +^{t}*t*, but not*f*(*xe*+^{t}*t*). - Given a group action of
*G*on*X*, we can define an induced action of*G*on the power set of*X*, by setting*g*⋅*U*= {*g*⋅*u*:*u*∈*U*} 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 geometries. - The quaternions with norm 1 (the versors), as a multiplicative group, act on
**R**^{3}: for any such quaternion*z*= cos*α*/2 +**v**sin*α*/2, the mapping*f*(**x**) =*z***x***z*^{∗}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.

## Group actions and groupoidsEdit

The notion of group action can be put in a broader context by using the *action groupoid* associated to the group action, thus allowing techniques from groupoid theory such as presentations and fibrations. 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 which is a *covering morphism of groupoids*. This allows a relation between such morphisms and covering maps in topology.

## Morphisms and isomorphisms between *G*-setsEdit

If *X* and *Y* are two *G*-sets, we define a *morphism* from *X* to *Y* to be a function *f* : *X* → *Y* such that *f*(*g*⋅*x*) = *g*⋅*f*(*x*) 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, and we call *f* an *isomorphism* and the two *G*-sets *X* and *Y* are called *isomorphic*; for all practical purposes, they are indistinguishable in this case.

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*G*×*S*, where*S*is some set and*G*acts on*G*×*S*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.the original action.)

With this notion of morphism, the collection of all *G*-sets forms a category; this category is a Grothendieck topos (in fact, assuming a classical metalogic, this topos will even be Boolean).

## Continuous group actionsEdit

One often considers *continuous group actions*: the group *G* is a topological group, *X* is a topological space, and the map *G* × *X* → *X* is continuous with respect to the product topology of *G* × *X*. 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. 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 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 topological space *X* arises in this manner: the quotient map *X* ↦ *X*/*G* 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 groupoid 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 (*x*, *y*) to (*y*, *x*).

An action of a group *G* on a locally compact space *X* is *cocompact* if there exists a compact subset *A* of *X* such that *GA* = *X*. 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 *G* × *X* → *X* × *X* that sends (*g*, *x*) ↦ (*g⋅x*, *x*) is a proper map.

### Strongly continuous group action and smooth pointsEdit

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 *g* ↦ *g*⋅*x* is continuous with respect to the respective topologies. Such an action induces an action on the space of continuous functions on *X* by defining (*g*⋅*f*)(*x*) = *f*(*g*^{−1}⋅*x*) 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.^{[11]}

The subspace of *smooth points* for the action is the subspace of *X* of points *x* such that *g* ↦ *g*⋅*x* is smooth, that is, it is continuous and all derivatives^{[where?]} are continuous.

## Variants and generalizationsEdit

One 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, one 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.

One 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 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 actions of topological groups on topological spaces, one also often considers smooth actions of Lie groups on smooth manifolds, regular actions of algebraic groups on algebraic varieties, and actions of group schemes on schemes. All of these are examples of group objects acting on objects of their respective category.

## See alsoEdit

## NotesEdit

**^**that is, the associated permutation representation is injective.

## CitationsEdit

**^**Eie & Chang (2010).*A Course on Abstract Algebra*. p. 144.**^**This is done, for example, by Smith (2008).*Introduction to abstract algebra*. p. 253.- ^
^{a}^{b}Thurston, William (1980),*The geometry and topology of three-manifolds*, Princeton lecture notes, p. 175 **^**Thurston 1980, p. 176.**^**tom Dieck, Tammo (1987),*Transformation groups*, de Gruyter Studies in Mathematics,**8**, Berlin: Walter de Gruyter & Co., p. 29, doi:10.1515/9783110858372.312, ISBN 978-3-11-009745-0, MR 0889050**^**Hatcher, Allen (2002).*Algebraic Topology*. Cambridge University Press. p. 72. ISBN 0-521-79540-0.**^**Procesi, Claudio (2007).*Lie Groups: An Approach through Invariants and Representations*. Springer Science & Business Media. p. 5. ISBN 9780387289298. Retrieved 23 February 2017.**^**M. Artin,*Algebra*, Proposition 6.4 on p. 179**^**Eie & Chang (2010).*A Course on Abstract Algebra*. p. 145.**^**Reid, Miles (2005).*Geometry and topology*. Cambridge, UK New York: Cambridge University Press. p. 170. ISBN 9780521613255.**^**Yuan, Qiaochu (27 February 2013). "wiki's definition of "strongly continuous group action" wrong?". Mathematics Stack Exchange. Retrieved 1 April 2013.

## ReferencesEdit

- Aschbacher, Michael (2000).
*Finite Group Theory*. Cambridge University Press. ISBN 978-0-521-78675-1. MR 1777008. - Brown, Ronald (2006).
*Topology and groupoids*, Booksurge PLC, ISBN 1-4196-2722-8. - Categories and groupoids, P.J. Higgins, downloadable reprint of van Nostrand Notes in Mathematics, 1971, which deal with applications of groupoids in group theory and topology.
- Dummit, David; Richard Foote (2004).
*Abstract Algebra*(3rd ed.). Wiley. ISBN 0-471-43334-9. - Eie, Minking; Chang, Shou-Te (2010).
*A Course on Abstract Algebra*. World Scientific. ISBN 978-981-4271-88-2. - Rotman, Joseph (1995).
*An Introduction to the Theory of Groups*. Graduate Texts in Mathematics**148**(4th ed.). Springer-Verlag. ISBN 0-387-94285-8. - Smith, Jonathan D.H. (2008).
*Introduction to abstract algebra*. Textbooks in mathematics. CRC Press. ISBN 978-1-4200-6371-4.

## External linksEdit

- Hazewinkel, Michiel, ed. (2001) [1994], "Action of a group on a manifold",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Weisstein, Eric W. "Group Action".
*MathWorld*.