Group Action

A group is said to act on a set when there is a map such that the following conditions hold for all elements .

1. where is the identity element of .

2. for all .

In this case, is called a transformation group, is a called a -set, and is called the group action.

In a group action, a group permutes the elements of . The identity does nothing, while a composition of actions corresponds to the action of the composition. For example, as illustrated above, the symmetric group acts on the digits 0 to 9 by permutations.

For a given , the set , where the group action moves , is called the group orbit of . The subgroup which fixes is the isotropy group of .

For example, the group acts on the real numbers by multiplication by . The identity leaves everything fixed, while sends to . Note that , which corresponds to . For , the orbit of is , and the isotropy subgroup is trivial, . The only group fixed point of this action is .

In a group representation, a group acts by invertible linear transformations of a vector space . In fact, a representation is a group homomorphism from to , the general linear group of . Some groups are described in a representation, such as the special linear group, although they may have different representations.

Historically, the first group action studied was the action of the Galois group on the roots of a polynomial. However, there are numerous examples and applications of group actions in many branches of mathematics, including algebra, topology, geometry, number theory, and analysis, as well as the sciences, including chemistry and physics.