faithful group action

Let A be a G-set, that is, a set acted upon by a group G with action ψ:G×AA. Then for any gG, the map mg:AA defined by


is a permutation of A (in other words, a bijective function from A to itself) and so an element of SA. We can even get an homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath from G to SA by the rule gmg.

If for any pair g,hG gh we have mgmh, in other words, the homomorphism gmg being injectivePlanetmathPlanetmath, we say that the action is faithfulPlanetmathPlanetmath.

Title faithful group action
Canonical name FaithfulGroupAction
Date of creation 2013-03-22 14:02:23
Last modified on 2013-03-22 14:02:23
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 8
Author rspuzio (6075)
Entry type Definition
Classification msc 16W22
Classification msc 20M30