Una acción de un grupo sobre un conjunto es una aplicación que cumple: 1. \n* donde es el elemento neutro del grupo. 2. \n* . Estas dos condiciones implican que, para cada elemento de , la aplicación es una función biyectiva. Otra posible definición, que se deriva de esto, es que una acción es un homomorfismo de grupos. .

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In mathematics, an action of a group is a way of interpreting the elements of the group as "acting" on 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. Some groups can be interpreted as acting on spaces in a canonical way. For example, the symmetric group of a finite set consists of all bijective transformations of that set; thus, applying any element of the permutation group to an element of the set will produce another element of the set. More generally, symmetry groups such as the homeomorphism group of a topological space or the general linear group of a vector space, as well as their subgroups, also admit canonical actions. For other groups, an interpretation of the group in terms of an action may have to be specified, either because the group does not act canonically on any space or because the canonical action is not the action of interest. For example, we can specify an action of the two-element cyclic group on the finite set by specifying that 0 (the identity element) sends , and that 1 sends . This action is not canonical. A common way of specifying non-canonical actions is to describe a homomorphism from a group G to the group of symmetries of a set X. The action of an element on a point is assumed to be identical to the action of its image on the point . The homomorphism is also frequently called the "action" of G, since specifying is equivalent to specifying an action. Thus, 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 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: \n* the identity element of G is assigned the identity transformation of X; \n* any product gk of two elements of G is assigned the composition of the permutations assigned to g and k. If X has additional structure, then is only called an action if for each , the permutation 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.

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