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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. A permutation representation of a group G is almost the same thing: formally it may be described as a group representation of G by permutation matrices, and is usually considered in the finite-dimensional case — it is the same as a group action of G on an ordered basis of a vector space.1 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.
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 → Sym(X), where Sym(X) denotes the group of all bijective maps from X to X.
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.
2 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 Sn 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.
- The symmetry group of any geometrical object acts on the set of points of that object.
- The automorphism group of a vector spaceThe fundamental concept in linear algebra is that of a vector space or linear space . This is a generalization of the set of all geometrical vectors and is used throughout modern mathematics. Formal definition A set V is a vector space over a field F (for (or graphGraph theory is the branch of mathematics that examines the properties of graphs . A graph with 6 vertices and 7 edges. Informally, a graph is a set of objects called vertices (or nodes) connected by links called edges (or arcs). Typically, a graph is dep, or group, or ring...) acts on the vector space (or set of verticesIn geometry, a vertex ( Latin: whirl, whirlpool; plural vertices is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). In graph theory, a graph describes a set of connections betw of the graph, or group, or ring...).
- The Lie groupIn mathematics, a Lie group (pronounced "lee", named after Sophus Lie) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. Lie groups are important in mathematical anas GL(n,R)Abstract algebra Algebra Linear algebra Lie groups In mathematics, the general linear group of degree n over a field F (such as R or C , written as GL ''n F , is the group of n ''n invertible matrices with entries from F with the group operation that of o, SL(n,R) and O(n,R)In mathematics, the orthogonal group of degree n over a field F (written as O n ''F ) is the group of n by n orthogonal matrices with entries from F with the group operation that of matrix multiplication. This is a subgroup of the general linear group GL act on Rn.
- 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.
