Group Action - Examples

Examples

• The trivial action of any group G on any set X is defined by g.x = x for all g in G and all x in X; that is, every group element induces the identity permutation on X.
• Every group G acts on G by left multiplication: g.x = gx for all g, x in G.
• Every group G acts on G by conjugation: g.x = gxg−1. An exponential notation is commonly used for the right-action variant: xg=g−1xg; it satisfies (xg)h=xgh.
• 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. It also acts on the set of faces or the set of edges of the 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 general linear group GL(n, R), special linear group SL(n, R), orthogonal group O(n, R), and special orthogonal group SO(n, R) are Lie groups which act on Rn. The group operations are given by multiplying the matrices from the groups with the vectors from Rn.
• The isometries of the plane act on the set of 2D images and patterns, such as wallpaper patterns. The definition can be made more precise by specifying what is meant by image or pattern; e.g., a function of position with values in a set of colors.
• More generally, if G acts on a set X, then G also acts in a natural way on the set of functions f:X→Y, using the rule (g.f)(x) = f(g−1.x) for every g in G, f:X→Y and x in X. Thus a group of bijections of a space induces a group action on "objects" defined on or in that space.
• 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) by time translation: 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.
• The additive group of the real numbers (R, +) acts on the set of real functions of a real variable in various ways, with (t.f)(x) equal to e.g. f(x + t), f(x) + t, f(xet), f(x)et, f(x + t)et, or f(xet) + t, but not f(xet + t).
• Given a group action of G on X, we can define an induced action of G on the power set of X, by setting g.U = {g.u : u ∈ U} for every subset U of X and every g in G. This is useful, for instance, in studying the action of the large Mathieu group on a 24-set and in studying symmetry in certain models of finite geometries.
• The quaternions with modulus 1, as a multiplicative group, act on R3: for any such quaternion, the mapping f(x) = z x z* is a counterclockwise rotation through an angle α about an axis v − z is the same rotation; see quaternions and spatial rotation.