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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 extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...
is a
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) wh ...
of a given
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
into the group of transformations of the space. Similarly, a group action on a
mathematical structure In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additional ...
is a group homomorphism of a group into the
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
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 Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
of a
polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...
acts on the vertices, the edges, and the
faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...
of the polyhedron. A group action on a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
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 Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
. The symmetric group acts on any
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
with elements by permuting the elements of the set. Although the group of all
permutation In mathematics, a permutation of a 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 to the act or proc ...
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 is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
.


Definition


Left group action

If is a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
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 from another group that allows one to define right action as a special case of left action. Monoids, groups, rings, and algebras can be viewed as categor ...
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, Cayley's theorem, named in honour of Arthur Cayley, states that every group is isomorphic to a subgroup of a symmetric group. More specifically, is isomorphic to a subgroup of the symmetric group \operatorname(G) whose eleme ...
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 (\mathbb Z/2\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 \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 ''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 group ''G'' is a non-em ...
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 (x_1, \ldots, x_n), (y_1, \ldots, y_n) \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 ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objec ...
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 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 by or Basic prop ...
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 matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
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 ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the genera ...
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. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit b ...
.


Primitive actions

The action of G on X is called ''primitive'' if there is no
partition Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
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 geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
and the action of G is by
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
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 In dynamical systems and ergodic theory, the concept of a wandering set formalizes a certain idea of movement and mixing. When a dynamical system has a wandering set of non-zero measure, then the system is a dissipative system. This is the opposit ...
''. 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 (x, y) = (2^n x, 2^ y) is wandering and free but not properly discontinuous. The action by
deck transformation 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 sp ...
s of the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
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, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...
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 (g, x) \mapsto (x, g\cdot x) is
proper Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
. This means that given compact sets K, K' the set of g \in G such that g \cdot K \cap K' \not= \emptyset is compact. In particular, this is equivalent to proper discontinuity when G is a
discrete group In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and on ...
. 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 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 along with the additio ...
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). One ma ...
, 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 Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
. There is a well-developed theory of
Lie group action In differential geometry, a Lie group action is a group action adapted to the smooth setting: G is a Lie group, M is a smooth manifold, and the action map is differentiable. __TOC__ Definition and first properties Let \sigma: G \times M \to M, ( ...
s, i.e. action which are smooth on the whole space.


Linear actions

If g acts by linear transformations on a
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...
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 Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
of ''X''. The associated
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...
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, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
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 In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology loo ...
and
group homology In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology lo ...
, 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 In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
group of ''G'' with coefficients in ''X'', and the higher cohomology groups are the
derived functor In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Motivation It was noted in vari ...
s 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 ''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, ''H'' 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 \to \operatorname(X), is given by the
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...
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 (g \cdot x) = g \cdot x. Applying g^ to both sides of this equality yields \left(g^ hg\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, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wor ...
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 In mathematics, the principal orbit type theorem states that compact Lie group acting smoothly on a connected differentiable manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. M ...
.


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(G) 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(g) = f(h) ''if and only if'' g and h lie in the same
coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
for the stabilizer subgroup G_x. Thus, the
fiber Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorporate ...
f^(\) 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 , G \cdot x, = \,:\,G_x= , G, / , 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 \bmod p orbits of length 1 which are ''G''-invariant elements. This result is especially useful since it can be employed for counting arguments (typically in situations where ''X'' is finite as well). : Example: We can use the orbit-stabilizer theorem to count the automorphisms of a
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
. Consider the
cubical graph In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...
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 mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
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, = , (G_1)\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\pi/3 which permutes 2,4,5 and 3,6,8, and fixes 1 and 7. Thus, \left, (G_1)\cdot2\ = 3. Applying the theorem a third time gives , \left(G_1\right)_2, = , \left(\left(G_1\right)_2\right)\cdot3, , \left(\left(G_1\right)_2\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(G_1\right)_2\right)\cdot3\ = 2. One also sees that \left(\left(G_1\right)_2\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 Burnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy–Frobenius lemma, the orbit-counting theorem, or the Lemma that is not Burnside's, is a result in group theory that is often useful in taking account of symmetry when ...
: , 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 In mathematics, the Burnside ring of a finite group is an algebraic construction that encodes the different ways the group can act on finite sets. The ideas were introduced by William Burnside at the end of the nineteenth century. The algebraic r ...
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 injection of each A_i into A, such that the images of these injections form a partition of A (th ...
, and multiplication to
Cartesian product In mathematics, specifically set theory, the Cartesian product of two 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 notation, that is : A\ti ...
.


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 In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to it ...
on ''X''. * In every group ''G'', left multiplication is an action of ''G'' on ''G'': for all ''g'', ''x'' in ''G''. This action is free and transitive (regular), and forms the basis of a rapid proof of
Cayley's theorem In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group is isomorphic to a subgroup of a symmetric group. More specifically, is isomorphic to a subgroup of the symmetric group \operatorname(G) whose eleme ...
- that every group is isomorphic to a subgroup of the symmetric group of permutations of the set ''G''. * In every group ''G'' with subgroup ''H'', left multiplication is an action of ''G'' on the set of cosets ''G/H'': for all ''g'',''a'' in ''G''. In particular if H contains no nontrivial normal subgroups of ''G'' this induces an isomorphism from ''G'' to a subgroup of the permutation group of degree '' : H'. * In every group ''G'',
conjugation Conjugation or conjugate may refer to: Linguistics * Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the chang ...
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 of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient ...
of a polyhedron acts on the set of vertices of that polyhedron. It also acts on the set of faces or the set of edges of the polyhedron. * The symmetry group of any geometrical object acts on the set of points of that object. * The automorphism group of a vector space (or
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
, 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 subgroup In mathematics, a Lie group (pronounced ) is a 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 along with the add ...
s (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 groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic grou ...
) are
Lie group In mathematics, a Lie group (pronounced ) is a 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 along with the additio ...
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 In mathematics, the affine group or general affine group of any affine space over a field is the group of all invertible affine transformations from the space into itself. It is a Lie group if is the real or complex field or quaternions. Relat ...
acts transitively on the points of an
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
, and the subgroup V of the affine group (that is, a vector space) has transitive and free (that is, ''regular'') action on these points; indeed this can be used to give a definition of an
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
. * The
projective linear group In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associate ...
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, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
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 Moebius, Möbius or Mobius may refer to: People * August Ferdinand Möbius (1790–1868), German mathematician and astronomer * Theodor Möbius (1821–1890), German philologist * Karl Möbius (1825–1908), German zoologist and ecologist * Paul ...
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 bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
of the plane act on the set of 2D images and patterns, such as
wallpaper pattern A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformati ...
s. 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) *Category (Kant) *Categories (Peirce) * ...
of ''G''-sets in which the objects are ''G''-sets and the
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
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 associated with the field extension. The study of field extensions and their relationship to the pol ...
of a
field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
''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 measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s acts on the
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
of "
well-behaved In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved. Th ...
" systems in
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
(and in more general
dynamical systems In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...
) by
time translation Time translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval. Time translation symmetry is the law that the laws of physics are unchanged (i ...
: if ''t'' is in R and ''x'' is in the phase space, then ''x'' describes a state of the system, and is defined to be the state of the system ''t'' seconds later if ''t'' is positive or −''t'' seconds ago if ''t'' is negative. *The additive group of the real numbers acts on the set of real functions of a real variable in various ways, with (''t''⋅''f'')(''x'') equal to, for example, , , , , , or , but not . *Given a group action of ''G'' on ''X'', we can define an induced action of ''G'' on the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of ''X'', by setting for every subset ''U'' of ''X'' and every ''g'' in ''G''. This is useful, for instance, in studying the action of the large
Mathieu group In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 obje ...
on a 24-set and in studying symmetry in certain models of finite geometries. * The
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s with
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
1 (the
versor In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Willi ...
s), 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 Unit quaternions, known as ''versors'', provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation abou ...
. 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 (id_X\times-g) (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 mathematics, specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed c ...
in the category of ''G''-sets.


Group actions and groupoids

The notion of group action can be encoded by the ''action
groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: *''Group'' with a partial functi ...
'' G'=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 map In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group ...
s'' or ''G-maps''. The composition of two morphisms is again a morphism. If a morphism ''f'' is bijective, then its inverse is also a morphism. In this case ''f'' is called an ''
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
'', and the two ''G''-sets ''X'' and ''Y'' are called ''isomorphic''; for all practical purposes, isomorphic ''G''-sets are indistinguishable. Some example isomorphisms: * Every regular ''G'' action is isomorphic to the action of ''G'' on ''G'' given by left multiplication. * Every free ''G'' action is isomorphic to , where ''S'' is some set and ''G'' acts on by left multiplication on the first coordinate. (''S'' can be taken to be the set of orbits ''X''/''G''.) * Every transitive ''G'' action is isomorphic to left multiplication by ''G'' on the set of left cosets of some subgroup ''H'' of ''G''. (''H'' can be taken to be the stabilizer group of any element of the original ''G''-set.) With this notion of morphism, the collection of all ''G''-sets forms a
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
; this category is a
Grothendieck topos In mathematics, a topos (, ; plural topoi or , or toposes) is a category (mathematics), category that behaves like the category of Sheaf (mathematics), sheaves of Set (mathematics), sets on a topological space (or more generally: on a Site (math ...
(in fact, assuming a classical metalogic, this topos will even be Boolean).


Variants and generalizations

We can also consider actions of
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
s on sets, by using the same two axioms as above. This does not define bijective maps and equivalence relations however. See
semigroup action In algebra and theoretical computer science, an action or act of a semigroup on a set is a rule which associates to each element of the semigroup a transformation of the set in such a way that the product of two elements of the semigroup (using t ...
. Instead of actions on sets, we can define actions of groups and monoids on objects of an arbitrary category: start with an object ''X'' of some category, and then define an action on ''X'' as a monoid homomorphism into the monoid of endomorphisms of ''X''. If ''X'' has an underlying set, then all definitions and facts stated above can be carried over. For example, if we take the category of vector spaces, we obtain
group representation In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to re ...
s in this fashion. We can view a group ''G'' as a category with a single object in which every morphism is invertible. A (left) group action is then nothing but a (covariant)
functor In mathematics, 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 In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition of m ...
, and a group representation is a functor from ''G'' to the
category of vector spaces In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ring o ...
. A morphism between G-sets is then a
natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
between the group action functors. In analogy, an action of a
groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: *''Group'' with a partial functi ...
is a functor from the groupoid to the category of sets or to some other category. In addition to continuous actions of topological groups on topological spaces, one also often considers smooth actions of Lie groups on
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold 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). One ma ...
s, regular actions of
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...
s on
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
, and
actions Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fi ...
of
group scheme In mathematics, a group scheme is a type of object from Algebraic geometry, algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of Scheme (mathematics), schemes, and they generalize algebraic groups, in ...
s on schemes. All of these are examples of
group object In category theory, a branch of mathematics, group objects are certain generalizations of groups that are built on more complicated structures than sets. A typical example of a group object is a topological group, a group whose underlying set is ...
s 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.


See also

*
Gain graph A gain graph is a graph whose edges are labelled "invertibly", or "orientably", by elements of a group ''G''. This means that, if an edge ''e'' in one direction has label ''g'' (a group element), then in the other direction it has label ''g''  ...
*
Group with operators In abstract algebra, a branch of mathematics, the algebraic structure group with operators or Ω-group can be viewed as a group with a set Ω that operates on the elements of the group in a special way. Groups with operators were extensively studi ...
*
Measurable group action In mathematics, a measurable acting group is a special group that Group action (mathematics), acts on some space in a way that is compatible with structures of measure theory. Measurable acting groups are found in the intersection of measure theor ...
*
Monoid action In algebra and theoretical computer science, an action or act of a semigroup on a set is a rule which associates to each element of the semigroup a transformation of the set in such a way that the product of two elements of the semigroup (using th ...


Notes


Citations


References

* * * * * * * * *


External links

* * {{Authority control Group theory Representation theory of groups Symmetry