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 
). We indicate this by
An important
special case
follows directly from the proof of :
.
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 .