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Group action
In mathematics, groups are often used to describe symmetries of objects. This is formalized by the notion of a group action: every element of the group "acts" like a bijective map (or "symmetry") on some set. In this case, the group is also called a transformation group of the set. A permutation representation of a group G is almost the same thing: formally it may be described as a group representation of G by permutation matrices, and is usually considered in the finite-dimensional case — it is the same as a group action of G on an ordered basis of a vector space.
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Definition
If G is a group and X is a set, then a (left) group action of G on X is a binary function G × X → X (where the image of g in G and x in X is written as g.x) which satisfies the following two axioms:
- g.(h.x) = (gh).x for all g, h in G and x in X.
- e.x = x for every x in X; here e denotes the identity element of G.
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. Therefore, one may alternatively and equivalently define a group action of G on X as a group homomorphism G → Sym(X), where Sym(X) denotes the group of all bijective maps from X to X.
If a group action G × X → X is given, we also say that G acts on the set X or X is a G-set.
In complete analogy, one can define a right group action of G on X as a function X × G → X by the two axioms (x.g).h = x.(gh) and x.e = x. In the sequel, we consider only left group actions.
Examples
- Every group G acts on G in two natural ways: g.x = (gx) for all x in G, or g.x = (gxg^{ -1}) for all x in G.
- The symmetric group S_{n} and its subgroups act on the set { 1, ... , n } by permuting its elements.
- The symmetry group of a polyhedron acts on the set of vertices of that 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, or group, or ring...) acts on the vector space (or set of vertices of the graph, or group, or ring...).
- The Lie groups GL(n,R), SL(n,R) and O(n,R) act on R^{n}.
- The Galois group of a field extension E/F acts on the bigger field E. So does every subgroup of the Galois group.
- The additive group of the real numbers (R, +) acts on the phase space of "well-behaved" systems in classical mechanics (and in more general dynamical systems): if t is in R and x is in the phase space, then x describes a state of the system, and t.x is defined to be the state of the system t seconds later if t is positive or -t seconds ago if t is negative.
Types of actions
The action of G on X is called
- transitive if for any two x, y in X there exists an g in G such that g.x = y;
- faithful (or effective) if for any two different g, h in G there exists an x in X such that g.x ≠ h.x;
- free if for any two different g, h in G and all x in X we have g.x ≠ h.x;
- regular (or simply transitive) if it is both transitive and free; this is equivalent to saying that for any two x, y in X there exists precisely one g in G such that g.x = y.
Every free action on a non-empty set is faithful. A group G acts faithfully on X iff the homomorphism G → Sym(X) has a trivial kernel. Thus, for a faithful action, G is isomorphic to a permutation group on X; specifically, G is isomorphic to its image in Sym(X).
The action of any group G on itself by left multiplication is regular, and thus faithful as well. Every group can, therefore, be embedded in the symmetric group on its own elements, Sym(G) — a result known as Cayley's theorem.
If G does not act faithfully on X, one can easily modify the group to obtain a faithful action. If we define N = {g in G : g.x = x for all x in X}, then N is a normal subgroup of G; indeed, it is the kernel of the homomorphism G → Sym(G). The factor group G/N acts faithfully on X by setting (gN).x = g.x. The original action of G on X is faithful if and only if N = {e}.
Orbits and stabilizers
Consider a group G acting on a set X. The orbit of a point x in X is the set of elements of X to which x can be moved by the elements of G. The orbit of x is denoted by Gx:
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 x ~ y iff 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 iff their orbits are the same, i.e. Gx = Gy.
The set of all orbits of X under the action of G is written as X/G.
If Y is a subset of X, we write GY for the set { g·y : y ∈ Y and g ∈ G}. We call the subset Y invariant under G if GY = Y (which is equivalent to GY ⊆ Y). In that case, G also operates on 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's 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.
For every x in X, we define the stabilizer subgroup of x (also called the isotropy group or little group) 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 G → Sym(X) is given by the intersection of the stabilizers G_{x} for all x in X.
Orbits and stabilizers are not unrelated. For a fixed x in X, consider the map from G to X given by g |-> g·x. The image of this map is the orbit of x and the coimage is the set of all left cosets of G_{x}. The standard quotient theorem of set theory then gives a natural bijection between G/G_{x} and Gx. Specifically, the bijection is given by hG_{x} |-> h·x. This result is known as the orbit-stabilizer theorem.
If G and X are finite then the orbit-stablilizer theorem, together with Lagrange's theorem, gives
This result is especially useful since it can be employed for counting arguments.
Note that if two elements x and y belong to the same orbit, then their stabilizer subgroups, G_{x} and G_{y}, are isomorphic. More precisely: if y = g·x, then G_{y} = gG_{x} g^{−1}.
A result closely related to the orbit-stabilizer theorem is Burnside's lemma:
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.
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 : 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.
If such a function 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:
- Every regular G action is isomorphic to the action of G on G given by left multiplication.
- Every free G action is isomorphic to G×S, where S is some set and G acts by left multiplication on the first coordinate.
- Every transitive G action is isomorphic to left multiplication by G on the set of left cosets of some subgroup H of G.
With this notion of morphism, the collection of all G-sets forms a category; this category is a topos.
Generalizations
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 above statements about isomorphisms for regular, free and transitive actions are no longer valid for continuous group actions.
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.
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 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. In analogy, an action of a groupoid is a functor from the groupoid to the category of sets or to some other category.