Double Cosets |

Corollary:IfGdenotes a finite group with subgroupsAandB,thenand if| AgB | =(| A | | B |)/(| A ÇgBg^{-1}|),Ddenotes a transversal of the setA \G/Bof(A,B)-double cosets, then| G | = å_{g Î D}| AgB | = å_{g Î D}(| A | | B |)/(| A ÇgBg^{-1}|).

The Cauchy-Frobenius Lemma yields the number of bilateral classes. In order to
evaluate it we have to calculate
the number * | G _{(a,b)} | *
of fixed points of

| {g | a=gbg^{-1}} |

which equals * | C _{G}(a) | = | C_{G}(b) | * if

Thus, by the Cauchy-Frobenius Lemma, we obtain

| U \\G | =(| G |)/(| U |)å_{(a,b) ÎU}(| C^{G}(a) ÇC^{G}(b) |)/(| C^{G}(a) |^{2}).

Corollary:IfUdenotes a subgroup ofG ´G,Gbeing a finite group, then the number of bilateral classes ofGwith respect toUisif(| G |)/(| U |)å_{g ÎC}(| C^{G}(g) ´C^{G}(g) ÇU |)/(| C^{G}(g) |),Cdenotes a transversal of the conjugacy classes of elements inG.In particular, the setofA \G/B:= {AgB | g ÎG }=(A ´B) \\G(A,B)-double cosets has the order| A \G/B | =(| G |)/(| A | | B |)å_{g ÎC}(| C^{G}(g) ÇA | | C^{G}(g) ÇB |)/(| C^{G}(g) |).

The main reason for the fact that double cosets show up nearly everywhere in the applications of group actions is the following one (which we immediately obtain from lemma):

Corollary:Ifis transitive and_{G}XUdenotes a subgroup ofG,then, for eachx ÎXwe have the natural bijectionIn particular, a transversalj:U \\X -> U \G/G_{x}:U(gx) -> UgG_{x}.Dof the set of double cosetsU \G/Gyields the following transversal of the set of orbits of_{x}U:T(U \\X):= {gx | g Î D }.

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

UNI-Graz | Institut für Mathematik |

UNI-Bayreuth | Lehrstuhl II für Mathematik |

Double Cosets |