on 
has first of all the following property which is
immediate from the two conditions mentioned in its definition:
This is the reason for the fact that 
induces several structures on
and , and
it is the close arithmetic and algebraic connection between these
structures which makes the concept of group action so efficient.
First of all the action induces the following equivalence
relation on (exercise 
):
The equivalence classes are called orbits , and
the orbit of 
will be indicated as follows:
As 
is an equivalence relation on , a transversal
of the orbits yields a set partition
of , i.e. a complete
dissection of into the pairwise disjoint and
nonempty subsets 
:
The set of orbits will be denoted by
In the case when both and are finite, we call the action a
finite action .
We notice that, according to , for each
finite -set , we may also assume without loss of generality
that is finite.
If G has exactly one orbit on , i.e. if and only if
, then we say that the action is transitive,
or
that acts transitively on .
According to an action of on yields a
partition of . It is trivial but very important
to notice that also the converse is true: Each
set partition of gives rise to an action of a certain group
on as follows. Let, for an index set 
, 
, 
, denote the
blocks of the set partition in question, i.e. the are nonempty, pairwise
disjoint, and their union is equal to . Then the following subgroup of the
symmetric group 
acts in a natural way on and has the as its
orbits:
Summarizing our considerations in two sentences, we have obtained:
.
Corollary
An action of a group on a set is equivalent to a permutation
representation of on and it yields a set partition of into
orbits. Conversely, each set partition of corresponds in a natural
way to an action of a certain subgroup of the symmetric group which has
the blocks of the partition as its orbits.
Exercises
E 
.
Prove that is in fact an equivalence relation.
