Similar 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/G_x).

A weaker concept is that of G- homomorphy.  

We shall write

_GX ~ _GY

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 G_x ÍG_y.

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

q(x)= q(gx)=g q(x).

On the other hand, if for each x ÎX there exist y ÎY such that G_x ÍG_y, 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 y_t ÎY such that G_t ÍG_yt. An easy check shows that

q:X -> Y :gt -> gy_t

is a well defined mapping and also a G-homomorphism.

Similar Actions