The
symmetry of a group
G with respect to an element
is the mapping
This is an old notion, which can be found in the book [
1]. And it is a very natural one, since
where
are the left translation, the right translation, and the inversion, respectively. Indeed, it follows from
that
, so
. Consequently,
. Similarly,
. Then
Various aspects of symmetries on groups had been studied in [
2].
Now let
G be a finite group and let
. An
r-coloring of
G is any mapping
. A coloring
of
G is
symmetric if there is
such that
for all
. That is, a coloring is symmetric if it is invariant under some symmetry. Define the equivalence relation ∼ on the set of all
r-colorings of
G by
That is, colorings are equivalent if one of them can be obtain from the other by a right translation.
Note that in the case of a finite cyclic group these notions have a very simple geometric illustration. Identifying with the vertices of a regular n-gon we obtain that a coloring is symmetric if it is invariant with respect to some mirror symmetry with an axis crossing the center of the polygon and one of its vertices.
![Symmetry 02 00069 i001]()
Colorings are equivalent if one of them can be obtained from the other by rotating about the center of the polygon.
![Symmetry 02 00069 i002]()
Obviously, the number of all
r-colorings of
G is
. Applying Burnside’s Lemma [
3, I, §3] shows that the number of equivalence classes of
r-colorings of
G is equal to
where
is the subgroup generated by
g. However, counting symmetric
r-colorings and equivalence classes of symmetric
r-colorings of
G turned out to be quite a difficult question.
Let
denote the set of symmetric
r-colorings of
G. In [
4] it was shown that if
G is Abelian, then
Here,
X runs over subgroups of
G,
Y over subgroups of
X,
is the Möbius function of the lattice of subgroups of
G, and
.
Given a finite partially ordered set, the
Möbius function is defined as follows:
See [
3, IV] for more information about the Möbius function.
In the case of
formulas
1,
2 were reduced to the following elementary ones [
4]:
If
n is odd then
If
, where
and
m is odd, then
As usual,
p denotes a prime number.
Recently, an approach for computing
and
in the case of an arbitrary finite group
G has been found [
5]. The approach is based on constructing the partially ordered set of so called optimal partitions of
G.
Given a partition
of
G, the
stabilizer and the
center of
are defined by
is a subgroup of
G and
is a union of left cosets of
G modulo
. Furthermore, if
, then
is also a union of right cosets of
G modulo
and for every
,
. We say that a partition
of
G is
optimal if
and for every partition
of
G with
and
, one has
. The latter means that every cell of
is contained in some cell of
, or equivalently, the equivalence corresponding to
is contained in that of
. The partially ordered set of optimal partitions of
G can be naturally identified with the partially ordered set of pairs
of subsets of
G such that
and
for some partition
of
G with
. For every partition
, we write
to denote the number of cells of
.
In [
5] it was shown that for every finite group
G and
,
where
P is the partially ordered set of optimal partitions of
G.
The partially ordered set of optimal partitions of G together with parameters , and can be constructed by starting with the finest optimal partition and using the following fact:
Let be an optimal partition of G and let . Let be the finest partition of G such that and , and let be the finest partition of G such that and . Then the partitions and are also optimal.
In this note we compute explicitly the numbers and where is the quaternion group.
First, we list all optimal partitions of Q together with parameters , and .
The finest partition
: , , , , .
, .
, , .
Then one partition
: , , , .
, .
, , .
Three partitions of the form
: , .
, .
, , .
And the coarsest partition
Next, we draw the partially ordered set
P of optimal partitions together with parameters
,
,
. The picture below shows also the values of the Möbius function of the form
.
![Symmetry 02 00069 i003]()
Finally, by formulas
3,
4, we obtain that
Thus, we have showed that
Proposition. For every , and .
In particular,
and
, while the number all 2-colorings of
Q is
and the number of equivalence classes of all 2-colorings of
Q is
We conclude this note with the list of all symmetric 2-colorings of Q, up to equivalence.