Definition. An action of a group on a set is a homomorphism from the group to the group of transformations of the set. The acting group may or may not be itself a transformation group.
If G is a group, X a set, and alpha an action of G on X, then one often adopts an operation-like notation
In this equivalent formulation one has two general rules:
(g_{1} g_{2}) . x = g_{1} . (g_{2} . x) for all g_{1}, g_{2} in G and all x in X.
1 . x = x for all x in X, where 1 denotes the identity element of G.
This course has been replete with examples of group actions. The examples include:
The action alpha of the additive group of R^{n} on itself by translations, i.e., (alpha(g))(x) = x + g for g and x in R^{n}. (Here one has alpha(g_{1} + g_{2}) = alpha(g_{1}) \circ alpha(g_{2}). )
The action mu of the group of orthogonal n \times n matrices on R^{n} by linear isometries, i.e., (mu(U))(x) = U x, for U an orthogonal matrix and x in R^{n}, when points of X = R^{n} are regarded as columns of length n.
The action of the isometry group of the plane on the plane.
New. The action of the affine group of the plane on the set of lines in the plane.
Definition. When a group acts on a set, the orbit of a point in the set is the collection of all points in that set that are obtained from the given point under the action. That is, if G acts on X and x is a given point of X, then the orbit of x is the set of all y in X with the property that y = g . x for one or more g in G.
Definition. The orbit space of a group action is the set of its orbits. Literally, if G acts on X, the orbit space is a set of subsets of X. However, one usually imagines the members of the orbit space as points in some abstract sense. For example, one regards the orbit space of the action of R^{n} on itself by translation as a single point since any point is in the orbit of the origin.
Find the smallest group of isometries of R^{2} that contains the six glide reflections formed using the six ways of composing three different reflections in the sides of a given equilateral triangle. (Note that it is not given that the reflections in the sides are included. Clearly, however, this group must be a subgroup of the smallest group containing the reflections in the three sides.)
Can every isometry of R^{3} be factored as the composition of plane reflections?
What is the orbit space for the action on R^{2} of the subgroup of the translation group consisting of all translations
What are the orbits for the action on R^{3} of the group of its rotations about the origin? And what is the orbit space of R^{3} for this action?