Double Cosets



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Double Cosets

. Corollary   If denotes a finite group with subgroups and then

and if denotes a transversal of the set of -double cosets, then

The Cauchy-Frobenius Lemma yields the number of bilateral classes. In order to evaluate it we have to calculate the number of fixed points of which is

Thus, by the Cauchy-Frobenius Lemma, we obtain

. Corollary   If denotes a subgroup of being a finite group, then the number of bilateral classes of with respect to is

if denotes a transversal of the conjugacy classes of elements in In particular, the set

of -double cosets has the order

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 gif):

. Corollary   If is transitive and denotes a subgroup of then, for each we have the natural bijection

In particular, a transversal of the set of double cosets yields the following transversal of the set of orbits of

Exercises

E .   Consider a -set , a normal subgroup , and the corresponding restriction . Check the following facts:



next up previous contents
Next: Action on k-subsets Up: Actions Previous: Bilateral classessymmetry



Herr Fripertinger
Sun Feb 05 18:28:26 MET 1995