Cosets |

Returning to the general case we first state the main
(and obvious) properties of the stabilizers of elements belonging to
the same orbit:
But the crucial point is the following
natural bijection between the orbit
of *x* and the set of left cosets of its stabilizer:

Lemma:The mappingG(x) -> G/Gis a bijection._{x}:gx -> gG_{x}

Proof: It is clear that, for a given *x ÎX*,
the following chain of equivalences holds:

gx=g'x iff g^{-1}g' ÎG_{x}iff g'G_{x}=gG_{x}.

Reading it from left to right we see that *gx -> gG _{x}*
defines a mapping,
reading it from right to left we obtain that it is injective. Furthermore
it is obvious that this mapping is also surjective.

This result shows in particular that *the length of the orbit is the
index of the stabilizer*, so that we obtain

Corollary:IfGis a finite group acting on the setX, then for eachx ÎXwe have| G(x) | = | G | / | G_{x}| .

An application to the examples given above yields:

Corollary:IfGis finite,g ÎG, andU £G, then the orders of the conjugacy classes of elements and of subgroups satisfy the following equations:| C^{G}(g) | = | G | / | C_{G}(g) | , and | [~U] | = | G | / | N_{G}(U) | .

harald.fripertinger "at" uni-graz.at | http://www-ang.kfunigraz.ac.at/~fripert/ |

UNI-Graz | Institut für Mathematik |

UNI-Bayreuth | Lehrstuhl II für Mathematik |

Cosets |