Group action


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 also acts on everything that is built on the 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. It allows one to identify many groups with subgroups of 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 G is a group and X is a set, then a group action of G on X is a function
that satisfies the following two axioms:
The group G is said to act on X. The set X is called a G-set.
From these two axioms, it follows that for every g in G, the function which maps x in X to is a bijective map from X to X. Therefore, one may alternatively define a group action of G on X as a group homomorphism from G into the symmetric group Sym of all bijections from X to X.

Right group action

In complete analogy, a right group action of G on X can be defined as a function satisfying the axioms
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 and is followed by g, while for a right action, g acts first and is followed by h. Because of the formula −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 it is sufficient to only consider left actions without any loss of generality.

Canonical maps

When there is a natural correspondence between the set of group elements and the set of space transformations, a group can be interpreted as acting on the space in a canonical way. For example, the symmetric group of a finite set consists of all bijective transformations of that set; thus, applying any element of the permutation group to an element of the set will produce another element of the set. More generally, symmetry groups such as the homeomorphism group of a topological space or the general linear group of a vector space, as well as their subgroups, also admit canonical actions. For other groups, an interpretation of the group in terms of an action may have to be specified, either because the group does not act canonically on any space or because the canonical action is not the action of interest. For example, we can specify an action of the two-element cyclic group C2 = on the finite set by specifying that 0 sends a ↦ a, b ↦ b, and c ↦ c, and that 1 sends a ↦ b, b ↦ a, and c ↦ c. This action is not canonical.

Types of actions

The action of G on X is called:
Furthermore, if G acts on a topological space X, then the action is:
If X is a non-zero module over a ring R and the action of G is R-linear then it is said to be
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 X under the action of G form a partition of X. The associated equivalence relation is defined by saying if and only if there exists a g in G with. 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,.
The group action is transitive if and only if it has exactly one orbit, that is, if there exists x in X with. This is the case if and only if for all x in X.
The set of all orbits of X under the action of G is written as X/G, and is called the quotient of the action. In geometric situations it may be called the ', while in algebraic situations it may be called the space of ', and written XG, by contrast with the invariants, 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.

Invariant subsets

If Y is a subset of X, one writes GY for the set. The subset Y is said invariant under G if . In that case, G also operates on Y by restricting the action to Y. The subset Y is called fixed under G if for all g in G and all y in Y. Every subset that is fixed under G is also invariant under G, but not conversely.
Every orbit is an invariant subset of X on which G acts transitively. The action of G on X is transitive if and only if all elements are equivalent, meaning that there is only one orbit.
A G-invariant element of X is such that for all. 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.

Fixed points and stabilizer subgroups

Given g in G and x in X with, 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 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,, 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.
Let x and y be two elements in X, and let g be a group element such that. Then the two stabilizer groups Gx and Gy are related by. Proof: by definition, if and only if. Applying g−1 to both sides of this equality yields ; that is,. An opposite inclusion follows similarly by taking and supposing.
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 of a subgroup of G. Let denote the conjugacy class of H. Then the orbit O has type if the stabilizer of some/any x in O belongs to. A maximal orbit type is often called a principal orbit type.

and Burnside's lemma

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 of this map is the orbit G·x. The condition for two elements to have the same image is
In other words, g and h lie in the same coset for the stabilizer subgroup. Thus the fiber of f over any y in G·x is such a coset, and clearly every such coset occurs as a fiber. Therefore f defines a bijection between the set of cosets for the stabilizer subgroup and the orbit G·x, which sends. 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.
This result is especially useful since it can be employed for counting arguments.
A result closely related to the orbit-stabilizer theorem is Burnside's lemma:
where Xg 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, and multiplication to Cartesian product.

Examples

The notion of group action can be put in a broader context by using the action groupoid associated to the group action, thus allowing techniques from groupoid theory such as presentations and fibrations. Further the stabilizers of the action are the vertex groups, and the orbits of the action are the components, of the action groupoid. For more details, see the book Topology and groupoids referenced below.
This action groupoid comes with a morphism p: G′ → G which is a covering morphism of groupoids. This allows a relation between such morphisms and covering maps in topology.

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:
With this notion of morphism, the collection of all G-sets forms a category; this category is a Grothendieck topos.

Continuous group actions

One often considers continuous group actions: the group G is a topological group, X is a topological space, and the map is continuous with respect to the product topology of. 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 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 to.
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. 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 that sends is a proper map.

Strongly continuous group action and smooth points

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 is continuous with respect to the respective topologies. Such an action induces an action on the space of continuous functions on X by defining 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 is smooth, that is, it is continuous and all derivatives are continuous.

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 group action is then nothing but a 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.

Citations