and if 
.
Corollary
If 
denotes a finite group with subgroups 
and 
then
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
if
of
.
Corollary
If 
denotes a subgroup of 
being a finite group, then the
number of bilateral classes of with respect to is
denotes a transversal of the conjugacy classes of elements in 
In particular,
the set
-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 
):
In particular, a transversal
.
Corollary
If 
is transitive and denotes a subgroup
of 
then, for each 
we have the natural bijection
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:
and any 
, the set
is also an orbit of
on .
on form a 
-set, in a natural way.
on 
are just the orbits of on
.
-orbits which belong to the same -orbit are of the same
order.