orbit-stabilizer theorem

Suppose that G is a group acting (http://planetmath.org/GroupAction) on a set X. For each xX, let Gx be the orbit of x, let Gx be the stabilizerMathworldPlanetmath of x, and let x be the set of left cosetsMathworldPlanetmath of Gx. Then for each xX the function f:Gxx defined by gxgGx is a bijection. In particular,




for all xX.

If yGx is such that y=g1x=g2x for some g1,g2G, then we have g2-1g1x=g2-1g2x=1x=x, and so g2-1g1Gx, and therefore g1Gx=g2Gx. This shows that f is well-defined.

It is clear that f is surjectivePlanetmathPlanetmath. If gGx=gGx, then g=gh for some hGx, and so gx=(gh)x=g(hx)=gx. Thus f is also injectivePlanetmathPlanetmath.

Title orbit-stabilizer theorem
Canonical name OrbitstabilizerTheorem
Date of creation 2013-03-22 12:23:10
Last modified on 2013-03-22 12:23:10
Owner yark (2760)
Last modified by yark (2760)
Numerical id 22
Author yark (2760)
Entry type TheoremMathworldPlanetmath
Classification msc 20M30