### Splitting Orbits

In the case when _{G}X is a finite action, we can apply the
sign map * e* to * bar (G)*, the permutation group induced by
*G* on *X*. Its kernel

* bar (G)*^{+}:= { bar (g) Îbar (G) | e( bar (g))
=1 }

is either * bar (G)* itself or a subgroup of index 2, as is easy to
see. Denoting its inverse image by

*G*^{+}:= {g ÎG | e( bar (g))=1 },

we obtain a useful interpretation of the *alternating sum* of fixed
point numbers:

**Lemma: **
*
For any finite action *_{G}X such that *G not = G*^{+}, the number of
orbits of *G* on *X* which split over *G*^{+} (i.e. which decompose into
more than one -- and hence into two -- *G*^{+}-orbits) is equal to
**(**1**)/(** | G | **)** å_{g ÎG} e( bar (g)) | X_{g}
| =**(**1**)/(** | bar (G) | **)** å_{ bar (g) Îbar (G)} e( bar (g)) | X_{ bar (g)} | .

Proof: As *G not =G*^{+}, and hence * | G*^{+}
| = | G | /2, we have

**(**1**)/(** | G | **)** å_{g ÎG} e( bar (g)) | X_{g}
| =**(**2**)/(** | G | **)** å_{g ÎG+} | X_{g} | -
**(**1**)/(** | G | **)** å_{g ÎG} | X_{g} |

*=***(**1**)/(** | G^{+} | **)** å_{g ÎG+} | X_{g} | - **(**1**)/(** | G | **)** å_{g ÎG} | X_{g} | = | G^{+} \\X | - | G \\X
| .

Each orbit of *G* on *X* is either a *G*^{+}-orbit or it splits into two
orbits of *G*^{+}, since * | G*^{+} | = | G | /2. Hence * | G*^{+} \\X
| - | G \\X | is the number of orbits which split over *G*^{+}.
Finally the stated identity is obtained by an application of the homomorphism
theorem.

**Corollary: **
*
In the case when **G not = G*^{+}, the number of *G*-orbits on *X* which
do not split over *G*^{+} is equal to
**(**1**)/(** | G | **)** å_{g ÎG}(1- e( bar (g))) | X_{g} | .

Note what this means. If *G* acts on a finite set *X* in such a way that
*G not =G*^{+}, then we can group the orbits of *G* on *X* into a set
of orbits which are also *G*^{+}-orbits. In the figure
we denote these orbits by the symbol * Ä*. The other *G*-orbits split
into two *G*^{+}-orbits, we indicate one of them by * Å*, the other
one by * OMINUS *, and call the pair * { Å, OMINUS }* an *enantiomeric pair* of *G*-orbits.
Hence the Lemma above gives
us the number of enantiomeric pairs of orbits, while corollary
yields the number of *selfenantiomeric*
orbits of *G* on *X*.
The elements *x ÎX* belonging to selfenantiomeric orbits are called
*achiral*
objects, while the others are
called *chiral*.
These notions of
*enantiomerism* and *chirality*
are taken from chemistry, where *G* is usually the
symmetry group of the molecule while *G*^{+} is its subgroup consisting
of the proper rotations. We call _{G}X a *chiral*
action if and only if *G not = G*^{+}.

Enantiomeric pairs and selfenantiomeric orbits

Using this notation we can now rephrase aboves lemma
and corollary
in the following way:

**Corollary: **
*
If *_{G}X is a finite chiral
action, then the number of
selfenantiomeric orbits of *G* on *X* is equal to
**(**1**)/(** | G | **)** å_{g ÎG}(1- e( bar (g))) | X_{g} | =
2 | G \\X | - | G^{+} \\X | ,

while the number of enantiomeric pairs of orbits is
**(**1**)/(** | G | **)** å_{g ÎG} e( bar (g)) | X_{g} | = | G^{+}
\\X | - | G \\X | .

The sign of a cyclic permutation is easy to obtain from the equation and the homomorphism property of the sign, described in the
corollary:

*
(i*_{1} ...i_{r}) ÎA_{n} iff r is odd.

But in fact we need not check the lengths of the cyclic factors of * p* since
an easy calculation shows (exercise) that,
in terms of the number *c( p)*
of cyclic factors of * p*, we have

*
e( p)=(-1)*^{n-c( p)}, if pÎS_{n} .

last changed: January 19, 2005