Effective Action

A group action G×X->X is effective if there are no trivial actions. In particular, this means that there is no element of the group (besides the identity element) which does nothing, leaving every point where it is. This can be expressed as  intersection _(x in X)G_x={e}, where G_x is the isotropy group at x and e is the identity of G.

It is possible for a Lie group G to have an effective action on a smaller dimensional space M. However,

 N(M)=max{dimG|G is a compact Lie group, acting effectively on M}

is finite, and is called the degree of symmetry of M.

See also

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

This entry contributed by Todd Rowland

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

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