Similar Actions



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Similar Actions

Under enumerative aspects is essentially the same as . This leads to the question of a suitable concept of morphism between actions of groups. To begin with, two actions will be called isomorphic iff they differ only by an isomorphism of the groups and a bijection between the sets which satisfy . In this case we shall write

in order to indicate the existence of such a pair of mappings. If we call and similar actions,   if and only if they are isomorphic by , where moreover , the identity mapping (cf. exercise gif). We indicate this by

An important special case follows directly from the proof of gif:

. Lemma   If is transitive then, for any , we have that

A weaker concept is that of - homomorphy. We shall write

if and only if there exists a mapping which is compatible with the action of Later on we shall see that the use of -homomorphisms is one of the most important tools in the constructive theory of discrete structures which can be defined as orbits of groups on finite sets. A characterization of -homomorphy gives

. Lemma   Two actions and are -homomorphic if and only if for each there exist such that

Proof: In the case when is a -homomorphism, then since, for each

On the other hand, if for each there exist such that we can construct a -homomorphism in the following way: Assume a transversal of the orbits, and choose, for each element of this transversal, an element such that An easy check shows that

is a well defined mapping and also a -homomorphism.

Exercises

E .   Check that the -isomorphy (and hence also the -similarity ) is an equivalence relation on group actions.

E .   Consider the following definition: We call actions and inner isomorphic if and only if there exists a pair such that and where is an inner automorphism  , which means that

for a suitable . Show that this equivalence relation has the same classes as .



next up previous contents
Next: Bilateral classessymmetry Up: Actions Previous: The Cauchy-Frobenius Lemma



Herr Fripertinger
Sun Feb 05 18:28:26 MET 1995