Paradigmatic Examples 2

. 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 :

and a natural action of on the set of orbits of :

Moreover the orbit 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:

. Lemma   The following mapping is a bijection:

if 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

E .   Check lemma .