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.
Table of contents |
2 Examples 3 Types of actions 4 Orbits and stabilizers 5 Morphisms and isomorphisms between G-sets 6 Generalizations |
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.
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 permutating 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;
- simply transitive if for any two x, y in X there exists precisely one 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;
Orbits and stabilizers
If we define N = {g in G : g.x = x for all x in X}, then N is a normal subgroup of G and the factor group G/N acts faithfully on X by setting (gN).x = g.x. The action of G on X is faithful if and only if N = {e}.
If Y is a subset of X, we write GY for the set { g.y : y in Y and g in 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.
Any operation of G on X defines an equivalence relation on X: two elements x and y are called equivalent if there exists a g in G with g.x = y. The equivalence class of x under this equivalence relation is given by the set Gx = { g.x : g in G } which is also called the orbit of x. The elements x and y are equivalent if and only if their orbits are the same: Gx = Gy. 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. The set of all orbits is written as X/G.
For every x in X, we define G_{x} = { g in G : g.x = x }. This is a subgroup of G, and it is called the stabilizer of x or isotropy subgroup at x. The action of G on X is free if and only if all stabilizers consist only of the identity element.
There is a natural bijection between the set of all left cosets of the subgroup G_{x} and the orbit of x, given by hG_{x} |-> h.x. Therefore, |Gx| = [G : G_{x}], and so
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. 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.
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.