Orbits

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= È_{t ÎT}G(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, X_i, i ÎI, denote the blocks of the set partition in question, i.e. the X_i are nonempty, pairwise disjoint, and their union is equal to X. Then the following subgroup of the symmetric group S_X acts in a natural way on X and has the X_i as its orbits:

  Å_i S_Xi := { pÎS_X | " i ÎI : pX_i= X_i }.

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 S_X which has the blocks of the partition as its orbits.

Orbits