In mathematics, a group action on a space is a group homomorphism 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 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 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 (finite-dimensional) vector space is called a representation of the group. It allows one to identify many groups with subgroups of GL(n, K), the group of the invertible matrices of dimension n over a field K.
The symmetric group Sn acts on any set with n 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.
If G is a group with identity element e, and X is a set, then a (left) group action α of G on X is a function
(with α(g, x) often shortened to gx or g ⋅ x when the action being considered is clear from context):
The group G is said to act on X (from the left). A set X together with an action of G is called a (left) G-set.
From these two axioms, it follows that for any fixed g in G, the function from X to itself which maps x to g ⋅ x is a bijection, with inverse bijection the corresponding map for g−1. Therefore, one may equivalently define a group action of G on X as a group homomorphism from G into the symmetric group Sym(X) of all bijections from X to itself.
(with α(x, g) often shortened to xg or x ⋅ g when the action being considered is clear from context)
The difference between left and right actions is in the order in which a product gh acts on x. For a left action, h acts first, followed by g second. For a right action, g acts first, followed by h second. Because of the formula (gh)−1 = h−1g−1, 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 G on X can be considered as a left action of its opposite group Gop on X.
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.
Consider a group G acting on a set X. The orbit of an element x in X is the set of elements in X to which x can be moved by the elements of G. The orbit of x is denoted by G⋅x:
The defining properties of a group guarantee that the set of orbits of (points x in) X under the action of G form a partition of X. The associated equivalence relation is defined by saying x ∼ y if and only if there exists a g in G with g⋅x = y. The orbits are then the equivalence classes under this relation; two elements x and y are equivalent if and only if their orbits are the same, that is, G⋅x = G⋅y.
The group action is transitive if and only if it has exactly one orbit, that is, if there exists x in X with G⋅x = X. This is the case if and only if G⋅x = X for all x in X (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 quotient of the action. In geometric situations it may be called the orbit space, while in algebraic situations it may be called the space of coinvariants, and written XG, by contrast with the invariants (fixed points), denoted XG: the coinvariants are a quotient while the invariants are a subset. The coinvariant terminology and notation are used particularly in group cohomology and group homology, which use the same superscript/subscript convention.
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 ∈ X such that g⋅x = x for all g ∈ G. The set of all such x is denoted XG 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 of G-invariants.
Given g in G and x in X with g⋅x = x, it is said that "x is a fixed point of g" or that "g fixes x". For every x in X, the stabilizer subgroup 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:
This is a subgroup of G, though typically not a normal one. The action of G on X is free if and only if all stabilizers are trivial. The kernel N of the homomorphism with the symmetric group, G → Sym(X), is given by the intersection of the stabilizers 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⋅x. Then the two stabilizer groups Gx and Gy are related by Gy = g Gx g−1. Proof: by definition, h ∈ Gy if and only if h⋅(g⋅x) = g⋅x. Applying g−1 to both sides of this equality yields (g−1hg)⋅x = x; that is, g−1hg ∈ Gx. An opposite inclusion follows similarly by taking h ∈ Gx and supposing x = g−1⋅y.
Orbits and stabilizers are closely related. For a fixed x in X, consider the map f:G → X given by g ↦ g·x. By definition the image f(G) of this map is the orbit G·x. The condition for two elements to have the same image is
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.
This result is especially useful since it can be employed for counting arguments (typically in situations where X is finite as well).
A result closely related to the orbit-stabilizer theorem is Burnside's lemma:
where Xg 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.
This action groupoid comes with a morphism p: G′ → G which is a covering morphism of groupoids. This allows a relation between such morphisms and covering maps in topology.
If X and Y are two G-sets, a morphism from X to Y is a function f : X → Y such that f(g⋅x) = g⋅f(x) for all g in G and all x in X. Morphisms of G-sets are also called equivariant maps or G-maps.
The composition of two morphisms is again a morphism. If a morphism f is bijective, then its inverse is also a morphism. 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.
One often considers continuous group actions: the group G is a topological group, X is a topological space, and the map G × X → X is continuous with respect to the product topology of G × X. The space X is also called a G-space in this case. This is indeed a generalization, since every group can be considered a topological group by using the discrete topology. All the concepts introduced above still work in this context, however we define morphisms between G-spaces to be continuous maps compatible with the action of G. The quotient X/G inherits the quotient topology from X, and is called the quotient space of the action. The above statements about isomorphisms for regular, free and transitive actions are no longer valid for continuous group actions.
If X is a regular covering space of another topological space Y, then the action of the deck transformation group on X is properly discontinuous as well as being free. Every free, properly discontinuous action of a group G on a path-connected topological space X arises in this manner: the quotient map X ↦ X/G is a regular covering map, and the deck transformation group is the given action of G on X. Furthermore, if X is simply connected, the fundamental group of X/G will be isomorphic to G.
These results have been generalized in the book Topology and Groupoids referenced below to obtain the fundamental groupoid of the orbit space of a discontinuous action of a discrete group on a Hausdorff space, as, under reasonable local conditions, the orbit groupoid of the fundamental groupoid of the space. This allows calculations such as the fundamental group of the symmetric square of a space X, namely the orbit space of the product of X with itself under the twist action of the cyclic group of order 2 sending (x, y) to (y, x).
An action of a group G on a locally compact space X is cocompact if there exists a compact subset A of X such that GA = X. For a properly discontinuous action, cocompactness is equivalent to compactness of the quotient space X/G.
The action of G on X is said to be proper if the mapping G × X → X × X that sends (g, x) ↦ (g⋅x, x) is a proper map.
A group action of a topological group G on a topological space X is said to be strongly continuous if for all x in X, the map g ↦ g⋅x is continuous with respect to the respective topologies. Such an action induces an action on the space of continuous functions on X by defining (g⋅f)(x) = f(g−1⋅x) for every g in G, f a continuous function on X, and x in X. Note that, while every continuous group action is strongly continuous, the converse is not in general true.
The subspace of smooth points for the action is the subspace of X of points x such that g ↦ g⋅x is smooth, that is, it is continuous and all derivatives[where?] are continuous.
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 from G to the category of sets, and a group representation is a functor from G to the category of vector spaces. A morphism between G-sets is then a natural transformation between the group action functors. In analogy, an action of a groupoid is a functor from the groupoid to the category of sets or to some other category.
In addition to continuous actions of topological groups on topological spaces, one also often considers smooth actions of Lie groups on smooth manifolds, regular actions of algebraic groups on algebraic varieties, and actions of group schemes on schemes. All of these are examples of group objects acting on objects of their respective category.