A group action of a group on a set 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 called a -set, and is called the group action.

In a group action, a group permutes the elements of . The identity element does nothing, while a composition of group 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 element leaves everything fixed, while sends to . Note that , which corresponds to . For , the group orbit of is , and the isotropy group is the trivial group, . 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 group representation is a group homomorphism from to , the general linear group of . Some groups are described in a group representation, such as the special linear group, although they may have different group representations.

An important example is the group 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.

See also

Action, Faithful Group Action, Free Action, Galois Group, Geometric Group Action, Group, Group Block, Group Orbit, Group Representation, Isotropy Group, Lie Group Quotient Space, Matrix Group, Primitive Group Action, Proper Group Action, Regular Group Action, Topological Group, Transitive Explore this topic in the MathWorld classroom

Portions of this entry contributed by Todd Rowland

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Cite this as:

Weisstein, Eric W., with contributions by Todd Rowland. "Group Action." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GroupAction.html

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