Group action

In mathematics, groups are often used to describe symmetries of objects. This is formalized in 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.

Definition

If G is a group and X is a set, then a 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) if the following two axioms are satisfied:

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 -> S(X), where S(X) denotes the group of all bijective maps from X to X.

Examples

Every permutation group S_n or A_n acts on the set { 1, ... , n }.
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 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.

Further definitions and facts

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. It is called free if g.x = x implies g = e for all g in G and x in X. The action is called faithful (or effective) if for any two g, h in G there exists an x in X such that g.x ≠ h.x.

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 quotient group G/N acts faithfully on X by setting (gN).x = g.x.

If Y is a subset of X, we write GY for the set { g.y : y in Y }. 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.

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.

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 Gx = Gy. There is a natural bijection between the set of all left cosets of the subgroup G_x and the orbit of x. Therefore, |Gx| = |G : G_x|, and so

|Gx| · |G_x| = |G|

This result, known as the Orbit-Stabilizer Theorem, is especially useful if G and X are finite, because then it can be employed for counting arguments. A related result is Burnside's Lemma:

r|G| = Σ_{g in G}|X_g|

where r is the number of orbits, and X_g is the set of points fixed by g. This result too is mainly of use when G and X are finite, when it can be used to determine the number of orbits.

If G acts on the set X, we also call X a G-set. The collection of all G-sets forms a category if we define a morphism from the G-set X to the G-set 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. The category of all G-sets is a topos.

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.

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.

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