Paradigmatic Examples

and notice:   If acts on and acts on , then , and act on as follows:

• , i.e. is mapped onto , where . The corresponding permutation group on will be denoted by .
• i.e. is mapped onto , where . The corresponding permutation group will be denoted by .
• , i.e. is mapped onto , where . The corresponding permutation group on will be denoted by , and it will be called the power group   of by .

and the multiplication is defined by

The actions and yield the following natural action of on :

The corresponding permutation group on will be denoted by , and it will be called the exponentiation group   of by . The orbits of and on will be called symmetry classes of mappings of mappings.

A few remarks concerning the wreath product are in order, they will in particular show that the actions of , , and on are restrictions of the action of on defined above. The reader is kindly asked carefully to check the following statements on wreath products :

. Lemma   The wreath product has the following properties:

• The identity element of is , where .
• If we define by , we get

  

• The normal subgroup

  

  is called the base group, and it is the direct product of copies of :

  

• The subgroup is a complement of , so that we have

  

• The diagonal

  

  satisfies

  

so that in fact the actions of , , and on introduced above are restrictions of the action of on .

In order to prepare later applications of such actions we mention that actions of direct products and of wreath products can be reformulated in terms of the respective factors of the direct and of the wreath product. Here is, to begin with, a lemma on actions of direct products, which is very easy to check (exercise ):

Exercises

E .   Show that is normal in and in , and that is not in general normal in . Check that the factor group is isomorphic to , while is isomorphic to . What does this mean, in the light of exercise , for the enumeration of the orbits of and ?