
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:
If a group action G × X > X is given, we also say that G acts on the set X or X is a Gset.
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.
The action of G on X is called
Every free action on a nonempty set is faithful. A group G that acts faithfully on a set X is isomorphic to a permutation group on X. An action is regular if and only if it is transitive and free.
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
If X and Y are two Gsets, 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 Gsets 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 Gsets forms a category; this category is a topos.
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 Gspace 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 Gspaces 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.
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