ExercisesActions of groupsThe Cauchy-Frobenius Lemma 2Similar Actions

Similar Actions

Under enumerative aspects GX is essentially the same as bar (G)X. 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 h:G simeq H of the groups and a bijection q:X -> Y between the sets which satisfy h(g) q(x) = q(gx). In this case we shall write
GX simeq HY,
in order to indicate the existence of such a pair of mappings. If G=H we call GX and GY similar actions, if and only if they are isomorphic by ( h, q), where moreover h= id G, the identity mapping (cf. exercise). We indicate this by
GX » GY.
An important special case follows directly from the proof:
Lemma: If GX is transitive then, for any x ÎX, we have that
GX » G(G/Gx).

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

if and only if there exists a mapping q:X -> Y which is compatible with the action of G: q(gx)=g q(x). Later on we shall see that the use of G-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 G-homomorphy gives
Lemma: Two actions GX and GY are G-homomorphic if and only if for each x ÎX there exist y ÎY such that Gx ÍGy.

Proof: In the case when q:X -> Y is a G-homomorphism, then Gx ÍG q(x) since, for each g ÎGx,

q(x)= q(gx)=g q(x).
On the other hand, if for each x ÎX there exist y ÎY such that Gx ÍGy, we can construct a G-homomorphism in the following way: Assume a transversal T(G \\X) of the orbits, and choose, for each element t of this transversal, an element yt ÎY such that Gt ÍGyt. An easy check shows that
q:X -> Y :gt -> gyt
is a well defined mapping and also a G-homomorphism.
last changed: August 28, 2001

ExercisesActions of groupsThe Cauchy-Frobenius Lemma 2Similar Actions