Group action

Given an equilateral triangle, the counterclockwise rotation by 120° around the center of the triangle maps every vertex of the triangle to another one. The cyclic group C3 consisting of the rotations by 0°, 120° and 240° acts on the set of the three vertices.

In mathematics, an action of a group is a way of interpreting the elements of the group as "acting" on some space in a way that preserves the structure of that space. Common examples of spaces that groups act on are sets, vector spaces, and topological spaces. Actions of groups on vector spaces are called representations of the group.

Some groups can be interpreted as acting on spaces 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 on the finite set by specifying that 0 (the identity element) sends , and that 1 sends . This action is not canonical.

A common way of specifying non-canonical actions is to describe a homomorphism from a group G to the group of symmetries of a set X. The action of an element on a point is assumed to be identical to the action of its image on the point . The homomorphism is also frequently called the "action" of G, since specifying is equivalent to specifying an action. Thus, if G is a group and X is a set, then an action of G on X may be formally defined as a group homomorphism from G to the symmetric group of X. The action assigns a permutation of X to each element of the group in such a way that:

If X has additional structure, then is only called an action if for each , the permutation preserves the structure of X.

The abstraction provided by group actions is a powerful one, because it allows geometrical ideas to be applied to more abstract objects. Many objects in mathematics have natural group actions defined on them. In particular, groups can act on other groups, or even on themselves. Because of this generality, the theory of group actions contains wide-reaching theorems, such as the orbit stabilizer theorem, which can be used to prove deep results in several fields.

Definition

If G is a group and X is a set, then a (left) group action φ of G on X is a function

that satisfies the following two axioms (where we denote φ(g, x) as g.x):[1]

Identity
e.x = x for all x in X. (Here, e denotes the identity element of the group G.)
Compatibility
(gh).x = g.(h.x) for all g, h in G and all x in X. (Here, gh denotes the result of applying the group operation of G to the elements g and h.)

The group G is said to act on X (on the left). The set X is called a (left) G-set.

From these two axioms, it follows that for every g in G, the function which maps x in X to g.x is a bijective map from X to X (its inverse being the function which maps x to g−1.x). Therefore, one may alternatively 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 X.[2]

In complete analogy, one can define a right group action of G on X as an operation X × GX mapping (x, g) to x.g and satisfying the two axioms:

Identity
x.e = x for all x in X.
Compatibility
x.(gh) = (x.g).h for all g, h in G and all x in X;

The difference between left and right actions is in the order in which a product like 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 (gh)−1 = h−1g−1, one can construct a left action from a right action by composing with the inverse operation of the group. Also, a right action of a group G on X is the same thing as a left action of its opposite group Gop on X. It is thus sufficient to only consider left actions without any loss of generality.

Types of actions

The action of G on X is called

Furthermore, if acts on a topological space , 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

Orbits and stabilizers

In the compound of five tetrahedra, the symmetry group is the (rotational) icosahedral group I of order 60, while the stabilizer of a single chosen tetrahedron is the (rotational) tetrahedral group T of order 12, and the orbit space I/T (of order 60/12 = 5) is naturally identified with the 5 tetrahedra – the coset gT corresponds to the tetrahedron to which g sends the chosen tetrahedron.

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 xy 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; i.e., G.x = G.y.

The group action is transitive if and only if it has only one orbit, i.e., 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.

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.

Invariant subsets

If Y is a subset of X, we write GY for the set { g.y : yY and gG}. We call the subset Y invariant under G if G.Y = Y (which is equivalent to G.YY). In that case, G also operates on Y by restricting the action to Y. The subset Y is called fixed under G if g.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 vice versa.

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 xX such that g.x = x for all gG. 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 group 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 g.x = x, we say x is a fixed point of g and g fixes x.

For every x in X, we define the stabilizer subgroup of G with respect to x (also called the isotropy group or little group[6]) as 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.

The above says that the stabilizers of elements in the same orbit are conjugate to each other. Thus, to each orbit, one can associate a conjugacy class of a subgroup of G (i.e., the set of all conjugates of the subgroup). Let denote the conjugacy class of H. Then one says that 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.

Orbit-stabilizer theorem and Burnside's lemma

Orbits and stabilizers are closely related. For a fixed x in X, consider the map from G to X given by gg.x for all gG. The image of this map is the orbit of x and the coimage is the set of all left cosets of Gx. The standard quotient theorem of set theory then gives a natural bijection between G/Gx and G.x. Specifically, the bijection is given by hGxh.x. This result is known as the orbit-stabilizer theorem. From a more categorical perspective, the orbit-stabilizer theorem comes from the fact that every G-set is a sum of quotients of the G-set G.

If G is finite then the orbit-stabilizer theorem, together with Lagrange's theorem, gives

This result is especially useful since it can be employed for counting arguments (typically in situations where X is finite as well).

Cubical graph with vertices labeled

Example: One can use the orbit-stabilizer theorem to count the automorphisms of a graph. Consider the cubical graph as pictured, and let denote its automorphism group. Then 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, we have that . Applying the theorem now to the stabilizer , we obtain . Any element of that fixes must send to either , or . There are such automorphisms; consider for example the map that transposes and , transposes and , and fixes the other vertices. Thus, . Applying the theorem a third time gives . Any element of that fixes and must send to either or , and one easily finds such automorphisms. Thus, . One also sees that consists only of the identity automorphism, as any element of fixing , and must also fix and consequently all other vertices. Combining the preceding calculations, we now obtain .

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.

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

Group actions and groupoids

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 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, we define a morphism from X to Y to be a function f : XY 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, and we call f an isomorphism and the two G-sets X and Y are called isomorphic; for all practical purposes, they are indistinguishable in this case.

Some example isomorphisms:

With this notion of morphism, the collection of all G-sets forms a category; this category is a Grothendieck topos (in fact, assuming a classical metalogic, this topos will even be Boolean).

Continuous group actions

One often considers continuous group actions: the group G is a topological group, X is a topological space, and the map G × XX 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 XX/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 × XX × X that sends (g, x) ↦ (g.x, x) 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 gg.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.[9]

The subspace of smooth points for the action is the subspace of X of points x such that gg.x is smooth; i.e., it is continuous and all derivatives are continuous.

Variants and generalizations

One 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, one 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.

One 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.

See also

Notes

  1. Eie & Chang (2010). A Course on Abstract Algebra. p. 144.
  2. This is done, e.g., by Smith (2008). Introduction to abstract algebra. p. 253.
  3. 1 2 Thurston, William (1980), The geometry and topology of three-manifolds, Princeton lecture notes, p. 175
  4. Thurston 1980, p. 176.
  5. tom Dieck, Tammo (1987), Transformation groups, de Gruyter Studies in Mathematics, 8, Berlin: Walter de Gruyter & Co., p. 29, ISBN 978-3-11-009745-0, MR 889050
  6. Procesi, Claudio (2007). Lie Groups: An Approach through Invariants and Representations. Springer Science & Business Media. p. 5. ISBN 9780387289298. Retrieved 23 February 2017.
  7. Eie & Chang (2010). A Course on Abstract Algebra. p. 145.
  8. Reid, Miles (2005). Geometry and topology. Cambridge, UK New York: Cambridge University Press. p. 170. ISBN 9780521613255.
  9. Yuan, Qiaochu (27 February 2013). "wiki's definition of "strongly continuous group action" wrong?". Mathematics Stack Exchange. Retrieved 1 April 2013.

References

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