What is "group action"

n. 1 (context group theory English) A mapping of a mathematical group to the permutations of an object that is compatible with the group operation. 2 (context sociology English) A situation in which a large number of agents take action simultaneously in order to achieve a common goal; their actions are usually coordinated.

n. action taken by a group of people

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 C = {0, 1} on the finite set {a, b, c} by specifying that 0 (the identity element) sends a ↦ a, b ↦ b, c ↦ c, and that 1 sends a ↦ b, b ↦ a, c ↦ c. This action is not canonical.

A common way of specifying non-canonical actions is to describe a homomorphism φ from a group G to the group of symmetries of a set X. The action of an element g ∈ G on a point x ∈ X is assumed to be identical to the action of its image φ(g) ∈ Sym(X) on the point x. The homomorphism φ is also frequently called the "action" of G, since specifying φ 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 φ 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:

• the identity element of G is assigned the identity transformation of X;
• 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 φ is only called an action if for each g ∈ G, the permutation φ(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.

In sociology, a group action is a situation in which a number of agents take action simultaneously in order to achieve a common goal; their actions are usually coordinated.

Group action will often take place when social agents realize they are more likely to achieve their goal when acting together rather than individually. Group action differs from group behaviours, which are uncoordinated, and also from mass actions, which are more limited in place.

Group action is more likely to occur when the individuals within the group feel a sense of unity with the group, even in personally costly actions.