Free Action

A group action G×X->X is called free if, for all x in X, gx=x implies g=I (i.e., only the identity element fixes any x). In other words, G×X->X is free if the map G×X->X×X sending (g,x) to (a(g,x),x) is injective, so that a(g,x)=x implies g=I for all g,x. This means that all stabilizers are trivial. A group with free action is said to act freely.

The basic example of a free group action is the action of a group on itself by left multiplication L:G×G->G. As long as the group has more than the identity element, there is no element h which satisfies gh=h for all g. An example of a free action which is not transitive is the action of S^1 on S^3 subset C^2 by e^(itheta)·(Z_1,Z_2)=(e^(itheta)Z_1,e^(itheta)Z_2), which defines the Hopf map.

See also

Effective Action, Group, Group Orbit, Group Representation, Isotropy Group, Lie Group Quotient Space, Matrix Group, Stabilizer, Topological Group, Transitive Group Action

This entry contributed by Todd Rowland

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

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