Orbits
An action of G on X has first of all the
following property which is
immediate from the two conditions mentioned in its definition:
gx=x' iff x=g-1x'.
This is the reason for the fact that GX induces
several structures on X and G, 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 X:
x ~G x' :iff $ g ÎG : x'=gx.
The equivalence classes are called orbits, and
the orbit of x ÎX
will be indicated as follows:
G(x) := {gx | g ÎG }.
As ~G is an equivalence relation on X, a transversal
T of the orbits yields a set partition
of X, i.e. a complete
dissection of X into the pairwise disjoint and
nonempty subsets G(t), t ÎT:
X= Èt ÎTG(t).
The set of orbits will be denoted by
G \\X := {G(t) | t Î T }.
In the case when both G and X are finite, we call the action a
finite action .
We notice that, according to
the representation of bar (G), for each
finite G-set X, we may also assume without loss of generality
that G is finite.
If G has exactly one orbit on X, i.e. if and only if
G \\X= {X }, then we say that the action is transitive,
or that G acts transitively on X.
According to the formula above
an action of G on X yields a
partition of X. It is trivial but very important
to notice that also the converse is true: Each
set partition of X gives rise to an action of a certain group G
on X as follows. Let, for an index set I,
Xi, i ÎI, denote the
blocks of the set partition in question,
i.e. the Xi are nonempty, pairwise
disjoint, and their union is equal to X.
Then the following subgroup of the
symmetric group SX acts in a natural
way on X and has the Xi as its orbits:
Åi SXi := { pÎSX | " i ÎI : pXi=
Xi }.
Summarizing our considerations in two sentences, we have obtained:
Corollary:
An action of a group G on a set X is equivalent to a permutation
representation of G on X and it yields a set partition of X into
orbits. Conversely, each set partition of X corresponds in a natural
way to an action of a certain subgroup of the symmetric group SX which has
the blocks of the partition as its orbits.
harald.fripertinger@kfunigraz.ac.at,
last changed: August 28, 2001
