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    Free Action

    Definition

    A group action G×X->X is called free if, for all x element X, g x = 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 g h = 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^(i θ)·(Z_1, Z_2) = (e^(i θ) Z_1, e^(i θ) Z_2), which defines the Hopf map.

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