Group Action

A group G is said to act on a set X when there is a map phi:G×X->X such that the following conditions hold for all elements x in X.

1. phi(e,x)=x where e is the identity element of G.

2. phi(g,phi(h,x))=phi(gh,x) for all g,h in G.

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

Group action of symmetric group

In a group action, a group permutes the elements of X. The identity does nothing, while a composition of actions corresponds to the action of the composition. For example, as illustrated above, the symmetric group S_(10) acts on the digits 0 to 9 by permutations.

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

For example, the group Z_2={[0],[1]} acts on the real numbers by multiplication by (-1)^n. The identity leaves everything fixed, while [1] sends x to -x. Note that [1]·[1]=[0], which corresponds to -(-x)=x. For x!=0, the orbit of x is {x,-x}, and the isotropy subgroup is trivial, {[0]}. The only group fixed point of this action is x=0.

In a group representation, a group acts by invertible linear transformations of a vector space V. In fact, a representation is a group homomorphism from G to GL(V), the general linear group of V. 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.

See also

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

This entry contributed by Todd Rowland

Explore with Wolfram|Alpha

Cite this as:

Rowland, Todd. "Group Action." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

Subject classifications