Properly discontinuous action

In mathematics, a group action is a formal way of interpreting the manner in which the elements of a group correspond to transformations of some space in a way that preserves the structure of that space. Common examples of spaces that groups act on are sets, vector spaces, and topological spaces. Actions of groups on vector spaces are called representations of the group. In mathematics, a group action is a formal way of interpreting the manner in which the elements of a group correspond to transformations of some space in a way that preserves the structure of that space. Common examples of spaces that groups act on are sets, vector spaces, and topological spaces. Actions of groups on vector spaces are called representations of the group. For a given (finite) set, the symmetric group is an abstraction used to describe the permutations of elements of that set. The concept of group action formalizes the relationship between the group and the permutations of the set. It relates each element of the group to a particular transformation. Subgroups of the symmetric group (including the symmetric group itself) are called permutation groups. A permutation representation of a group G is a representation of G as a group of permutations of some set, and may be described as a group representation of G by permutation matrices. To include non-finite cases, the concept of permutation is generalized as a bijective transformation. The bijective transformations of a set form a group, whose subgroups are transformation groups. An example is the group of linear transformations that act on a vector space. Group action is an extension to the notion of symmetric group in which every element of the group acts as a bijective transformation on the given set, without being identified with that transformation. This allows for a more comprehensive description of transformations (such as the symmetries of a polyhedron), by allowing the same group to act on several different sets of features (such as the set of vertices, the set of edges or the set of faces of the polyhedron). If G is a group and X is a set, then an action of G on X may be formally defined as a group homomorphism φ {displaystyle varphi } from G to the symmetric group of X. The action assigns a permutation of X to each element of the group in such a way that: If X has additional structure, then φ {displaystyle varphi } is only called an action if for each g ∈ G {displaystyle gin G} , the permutation φ ( g ) {displaystyle varphi (g)} preserves the structure of X. The abstraction provided by group actions is a powerful one, because it allows geometrical ideas to be applied to more abstract objects. Many objects in mathematics have natural group actions defined on them. In particular, groups can act on other groups, or even on themselves. Because of this generality, the theory of group actions contains wide-reaching theorems, such as the orbit-stabilizer theorem, which can be used to prove deep results in several fields. If G is a group and X is a set, then a (left) group action φ of G on X is a function that satisfies the following two axioms (where we denote φ(g, x) as g⋅x):