Group action (mathematics)

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OR:

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 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 :$\alpha\colon G \times X \to X,$ 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 :$\alpha\colon X \times G \to X,$ 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 its
opposite 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 $g \cdot x = x$ for all $x \in X$ 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 $g \cdot x = x$ for some $x \in X$ 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 implies
Cayley'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 $\left(\mathbb Z/2\mathbb Z\right)^n$ (of cardinality $2^n$) acts faithfully on a set of size $2n$. This is not always the case, for example the cyclic group $\mathbb Z/2^n\mathbb Z$ 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 $A_5 \times \mathbb Z/2\mathbb Z$ and the cyclic group $\mathbb Z / 120\mathbb Z$. 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 $x, y \in X$ there exists a $g \in G$ so that $g \cdot x = y$. The action is ' (or ''sharply transitive'', or ') if it is both transitive and free. This means that given $x, y \in X$ 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 a
principal 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 $n \ge 1$, the action is if $X$ has at least $n$ elements, and for any pair of $n$-tuples $\left(x_1, \ldots, x_n\right), \left(y_1, \ldots, y_n\right) \in X^n$ with pairwise distinct entries (that is $x_i \not=x_j$, $y_i \not=y_j$ when $i \not= j$) there exists a $g \in G$ such that $g \cdot x_i = y_i$ for $i=1,\ldots,n$. 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 the
alternating 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 $\left(n-2\right)$-transitive but not $\left(n-1\right)$-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 \setminus \$ 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 a
topological 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 $x \in X$ has a neighbourhood $U$ such that there are only finitely many $g \in G$ with $g\cdot U \cap U \not= \emptyset$. More generally, a point $x \in X$ is called a point of discontinuity for the action of $G$ if there is an open subset $U \ni x$ such that there are only finitely many $g \in G$ with $g\cdot U \cap U \not= \emptyset$. The ''domain of discontinuity'' of the action is the set of all points of discontinuity. Equivalently it is the largest $G$-stable open subset $\Omega \subset X$ such that the action of $G$ on $\Omega$ is wandering. In a dynamical context this is also called '' wandering set''. The action is ''properly discontinuous'' if for every compact subset $K \subset X$ there are finitely many $g \in G$ such that $g \cdot K \cap K \not= \emptyset$. This is strictly stronger than wandering; for instance the action of $\mathbb Z$ on $\mathbb^2 \setminus \$ given by $n\cdot \left(x, y\right) = \left(2^n x, 2^ y\right)$ 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 $x \in X$ has a neighbourhood $U$ such that $g \cdot U \cap U = \emptyset$ for every $g \in G \setminus \$. 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 $A \subset X$ such that $X = G \cdot A$. For a properly discontinuous action, cocompactness is equivalent to compactness of the quotient space $G \backslash X$.

## Actions of topological groups

Now assume $G$ is a
topological 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 $G \times X \to X$ is continuous for the product topology. The action is said to be ' if the map $G \times X \to X \times X$ defined by $\left(g, x\right) \mapsto \left(x, g\cdot x\right)$ is proper. This means that given compact sets $K, K\text{'}$ the set of $g \in G$ such that $g \cdot K \cap K\text{'} \not= \emptyset$ 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 $g \cdot x \not= x$ for all $x \in X$ and $g \in U \setminus \$. The action is said to be ''strongly continuous'' if the orbital map $g \mapsto g \cdot x$ is continuous for every $x \in X$. 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 $x \in X$ such that the map $x \mapsto g \cdot x$ 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 $G \cdot x$: $G \cdot x = \.$ 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 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 $x \sim y$
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 $g \cdot x = y.$ 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, $G \cdot x = G \cdot y.$ The group action is transitive if and only if it has exactly one orbit, that is, if there exists ''x'' in ''X'' with $G \cdot x = X.$ This is the case if and only if $G \cdot x = X$ 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 ''XG'': 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.

## Invariant subsets

If ''Y'' is a
subset 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 $G \cdot Y$ denotes the set $\.$ The subset ''Y'' is said to be ''invariant under G'' if $G \cdot Y = Y$ (which is equivalent to $G \cdot Y \subseteq Y$). In that case, ''G'' also operates on ''Y'' by restricting the action to ''Y''. The subset ''Y'' is called ''fixed under G'' if $g \cdot y = y$ 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 $x \in X$ such that $g \cdot x = x$ for all $g \in G.$ The set of all such ''x'' is denoted $X^G$ and called the ''G-invariants'' of ''X''. When ''X'' is a ''G''-module, ''XG'' is the zeroth cohomology group of ''G'' with coefficients in ''X'', and the higher cohomology groups are the derived functors of the
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 ...
of ''G''-invariants.

## Fixed points and stabilizer subgroups

Given ''g'' in ''G'' and ''x'' in ''X'' with $g \cdot x = x,$ 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'': $G_x = \.$ This is a
subgroup 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, $G \to \operatorname\left(X\right),$ is given by the intersection of the stabilizers ''Gx'' 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 $y = g \cdot x.$ Then the two stabilizer groups $G_x$ and $G_y$ are related by $G_y = g G_x g^.$ Proof: by definition, $h \in G_y$ if and only if $h \cdot \left(g \cdot x\right) = g \cdot x.$ Applying $g^$ to both sides of this equality yields $\left\left(g^ hg\right\right) \cdot x = x;$ that is, $g^ h g \in G_x.$ An opposite inclusion follows similarly by taking $h \in G_x$ and supposing $x = g^ \cdot y.$ 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
conjugacy 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 $\left(H\right)$ denote the conjugacy class of ''H''. Then the orbit ''O'' has type $\left(H\right)$ if the stabilizer $G_x$ of some/any ''x'' in ''O'' belongs to $\left(H\right)$. 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 $f : G \to X$ given by $g \mapsto g \cdot x.$ By definition the image $f\left(G\right)$ of this map is the orbit $G \cdot x.$ The condition for two elements to have the same image is $f(g)=f(h) \iff g\cdot x=h \cdot x \iff g^h \cdot x=x \iff g^h \in G_x \iff h \in gG_x.$ In other words, $f\left(g\right) = f\left(h\right)$ ''if and only if'' $g$ and $h$ lie in the same coset for the stabilizer subgroup $G_x$. Thus, the
fiber 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 ...
$f^\left(\\right)$ 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 $G \cdot x,$ which sends $gG_x \mapsto g \cdot x$. This result is known as the ''orbit-stabilizer theorem''. If ''G'' is finite then the orbit-stabilizer theorem, together with Lagrange's theorem, gives 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 $k \bmod p$ 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, $, G, = , G\cdot1, , G_1, = 8, G_1, .$ Applying the theorem now to the stabilizer $G_1,$ we can obtain $, G_1, = , \left(G_1\right)\cdot2, , \left(G_1\right)_2, .$ 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 $2\pi/3$ which permutes 2,4,5 and 3,6,8, and fixes 1 and 7. Thus, $\left, \left(G_1\right)\cdot2\ = 3.$ Applying the theorem a third time gives $, \left\left(G_1\right\right)_2, = , \left\left(\left\left(G_1\right\right)_2\right\right)\cdot3, , \left\left(\left\left(G_1\right\right)_2\right\right)_3, .$ 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 $\left, \left\left(\left\left(G_1\right\right)_2\right\right)\cdot3\ = 2$. One also sees that $\left\left(\left\left(G_1\right\right)_2\right\right)_3$ 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 $, G, = 8\cdot3\cdot2\cdot1 = 48.$ A result closely related to the orbit-stabilizer theorem is Burnside's lemma: $, X/G, =\frac\sum_ , X^g, ,$ where ''X''''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
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 of
Cayley'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 S''n'' and its subgroups act on the set by permuting its elements * The
symmetry 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''''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
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 ...
Pn(''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
cross 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 R3: 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 $\alpha:X\times G\to Y$, and with left ''G''-action given by $g\cdot\alpha=\alpha\circ \left(id_X\times-g\right)$ (where "$-g$" indicates right multiplication by $g$). This ''G''-set has the property that its fixed points correspond to equivariant maps $X\to Y$; more generally, it is an exponential object in the category of ''G''-sets.

# Group actions and groupoids

The notion of group action can be encoded by the ''action groupoid'' $G\text{'}=G \ltimes X$ 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

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.

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

# References

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