. 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 ):
. 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: