### Similar Actions

Under enumerative aspects _{G}X 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

_{G}X simeq _{H}Y,

in order to indicate the existence of such a pair of mappings. If
*G=H* we call _{G}X and _{G}Y *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

_{G}X » _{G}Y.

An important special case
follows directly from the proof:

**Lemma: **
*
If *_{G}X is transitive then, for any *x ÎX*,
we have that
_{G}X » _{G}(G/G_{x}).

A weaker concept is that of *G*- *homomorphy.*

We shall write

_{G}X ~ _{G}Y

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 *_{G}X and _{G}Y 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.

last changed: January 19, 2005