and a natural action of
Moreover the orbit of
.
Lemma
In each case when a direct product 
acts on a set 
,
we obtain both a natural action of 
on the set of orbits of 
:
on the set of orbits of :
under is the set consisting of
the orbits of on that form 
, while the orbit of
under is the set consisting of the orbits of
on that form , and therefore the following
identity holds:
In particular each action of the form
can be considered as an action of on 
or
as an action of on 
.
The corresponding result on wreath products is due to W. Lehmann, and it reads as follows:
if
.
Lemma
The following mapping is a bijection:
is defined by 
In particular,
Proof: It is easy to see that 
is well defined.
In order to prove that is injective assume
, so that there exist
such that 
But this implies
for each 
Therefore there must exist, for each 
such that 
Summing up there exist
for which 
, and so
In order to show that is surjective assume that
Defining 
in such a way that
, say 
for all 
, then, for 
defined as above,
which gives
and it completes the proof.
Exercises
.
