Group action

In mathematics, an action of a group on a set is, loosely speaking, an operation that takes an element of and an element of and produces another element of More formally, it is a group homomorphism from to the automorphism group of . One says that acts on
Many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles.
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 and also on the figures drawn in it; in particular, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron.
A group action on a vector space 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 general linear group, the group of the invertible matrices 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 permutations 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.

Definition

Left group action

If is a group with identity element, and is a set, then a group action of on is a function
that satisfies the following two axioms:
for all and in and all in.
The group is then said to act on . A set together with an action of is called a -set.
It can be notationally convenient to curry the action, so that, instead, one has a collection of transformations, with one transformation for each group element. The identity and compatibility relations then read
and
The second axiom states that the function composition is compatible with the group multiplication; they form a commutative diagram. This axiom can be shortened even further, and written as.
With the above understanding, it is very common to avoid writing entirely, and to replace it with either a dot, or with nothing at all. Thus, can be shortened to or, especially when the action is clear from context. The axioms are then
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
that satisfies the analogous axioms:
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 on. Thus, for establishing general properties of a single group action, it suffices to consider only left actions.

Notable properties of actions

Let be a group acting on a set. The action is called ' or ' if for all implies that. Equivalently, the homomorphism from to the group of bijections of corresponding to the action is injective.
The action is called if the statement that for some already implies that. In other words, no non-trivial element of fixes a point of. 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 that any group can be embedded in a symmetric group. A finite group may act faithfully on a set of size much smaller than its cardinality. For instance the abelian 2-group acts faithfully on a set of size. This is not always the case, for example the cyclic group cannot act faithfully on a set of size less than.
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, the icosahedral group and the cyclic group. The smallest sets on which faithful actions can be defined for these groups are of size 5, 7, and 16 respectively.

Transitivity properties

The action of on is called ' if for any two points there exists a so that.
The action is ' if it is both [|transitive] and free. This means that given there is exactly one such that. If is acted upon simply transitively by a group then it is called a principal homogeneous space for or a -torsor.
For an integer, the action is if has at least elements, and for any pair of -tuples with pairwise distinct entries there exists a such that for. In other words, the action on the subset of of tuples without repeated entries is transitive. For this is often called double, respectively triple, transitivity. The class of 2-transitive groups and more generally multiply transitive groups is well-studied in finite group theory.
An action is when the action on tuples without repeated entries in is sharply transitive.

Examples

The action of the symmetric group of is transitive, in fact -transitive for any up to the cardinality of. If has cardinality, the action of the alternating group is -transitive but not -transitive.
The action of the general linear group of a vector space on the set of non-zero vectors is transitive, but not 2-transitive. The action of the orthogonal group of a Euclidean space is not transitive on nonzero vectors but it is on the unit sphere.

Primitive actions

The action of on is called primitive if there is no partition of preserved by all elements of apart from the trivial partitions.

Topological properties

Assume that is a topological space and the action of is by homeomorphisms.
The action is wandering if every has a neighbourhood such that there are only finitely many with.
More generally, a point is called a point of discontinuity for the action of if there is an open subset such that there are only finitely many with. The domain of discontinuity of the action is the set of all points of discontinuity. Equivalently it is the largest -stable open subset such that the action of on is wandering. In a dynamical context this is also called a wandering set.
The action is properly discontinuous if for every compact subset there are only finitely many such that. This is strictly stronger than wandering; for instance the action of on given by is wandering and free but not properly discontinuous.
The action by deck transformations of the fundamental group of a locally simply connected space on a universal cover is wandering and free. Such actions can be characterized by the following property: every has a neighbourhood such that for every. 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 on a locally compact space is called cocompact if there exists a compact subset such that. For a properly discontinuous action, cocompactness is equivalent to compactness of the quotient space.

Actions of topological groups

Now assume is a topological group and a topological space on which it acts by homeomorphisms. The action is said to be continuous if the map is continuous for the product topology.
The action is said to be if the map defined by is proper. This means that given compact sets the set of such that is compact. In particular, this is equivalent to proper discontinuity if is a discrete group.
It is said to be locally free if there exists a neighbourhood of such that for all and.
The action is said to be strongly continuous if the orbital map is continuous for every. Contrary to what the name suggests, this is a weaker property than continuity of the action.
If is a Lie group and a differentiable manifold, then the subspace of smooth points for the action is the set of points such that the map 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 acts by linear transformations on a module over a commutative ring, the action is said to be irreducible if there are no proper nonzero -invariant submodules. It is said to be semisimple if it decomposes as a direct sum of irreducible actions.

Orbits and stabilizers

Consider a group acting on a set. The ' of an element in is the set of elements in to which can be moved by the elements of. The orbit of is denoted by :
The defining properties of a group guarantee that the set of orbits of under the action of form a partition of. The associated equivalence relation is defined by saying if and only if there exists a in with. The orbits are then the equivalence classes under this relation; two elements and are equivalent if and only if their orbits are the same, that is,.
The group action is transitive if and only if it has exactly one orbit, that is, if there exists in with. This is the case if and only if for in .
The set of all orbits of under the action of is written as , 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, by contrast with the invariants, denoted : 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.