Faithful Group Action

A group action phi:G×X->X is called faithful if there are no group elements g (except the identity element) such that gx=x for all x in X. Equivalently, the map phi induces an injection of G into the symmetric group S_X. So G can be identified with a permutation subgroup.

Most actions that arise naturally are faithful. An example of an action which is not faithful is the action e^(i(x+y)) of G=R^2={(x,y)} on X=S^1={e^(itheta)}, i.e., phi(x,y,e^(itheta))=e^(i(theta+x+y)).

See also

Ado's Theorem, Effective Action, Free Action, Group, Group Orbit, Iwasawa's Theorem, Lie Group Quotient Space, Transitive

This entry contributed by Todd Rowland

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Cite this as:

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

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