For a given
combinatorial map its edge rotation is equal to.
Let us
remind that permutations and are called conjugated with respect to permutation if. We see that in combinatorial map inner edge rotation and
inner edge rotation are conjugated with respect to vertex rotation.
Are there
other maps with the same edge rotation [within the class of maps with fixed
inner edge rotation]? Yes, each permutation with respect which and are conjugated fits
for vertex rotation of such map.
Let inner
edge rotation be fixed. All such classes form a class of maps:
.
Let us
define class of combinatorial maps that contain maps with fixed edge rotation:
.
For
different values of edge rotation class have subclasses. Between these classes one class is special, namely, that which has, i.e., for the members of this class edge rotation and inner
edge rotation coincided:
.
Maps of this class are called selfconjugate
maps. Thus, is the class of selfconjugate maps. This class is not empty; it
contains map . Indeed, . Thus, map with only isolated edges is example of
selfconjugate maps. We shall see further other examples too.
In order to
learn to consider maps of , let us find what edge rotation has multiplication of two
maps.
Theorem 1. For
two maps there holds: .
Proof:
.
Let us
define class where is fixed:
,
i.e., contain all maps from multiplied from left
with a fixed map . From theorem 1 we have that , where . Let us prove that equality holds. Let us first prove that is a group.
Theorem 2 .
is a group.
Proof: Group
operation is, of course, multiplication of permutations. Let us show that is closed with respect
to multiplication of permutations. If then . Class is closed with respect to reversion operation too: .Class contains a map that
corresponds to identity permutation.
Thus, all group’s requirements satisfies, and is a group.
Let us note
that is class of maps with fixed and in the same time
it is a group [or isomorphic to] that is called symmetry group. [Mostly it is
designated as ; in our case we use designation ]. Of course, is subgroup of group . Let us prove coincidence of two classes.
Theorem 3. . (.)
Proof: If , then is a left coset of equal to and theorem is proved.
Let and is a map, that holds, i.e., is left coset of and is one of elements of
class : and according to theorem 1. If , then holds
.
Besides the
theorem, we got that class is left coset of the
groupin the group . Let us fix this as corollary.
Corollary
1. In the group left cosets to the
subgroup are classes with fixed
edge rotation.
Thus, arbitrary
combinatorial map with edge rotation
equal to belongs to coset to .
Let us
consider some properties of class .
Lemma 1. holds iff .
Proof:
.
Let be orbit of . We call orbit selfconjugate with respect to orbit (with respect to) if . If in a map each orbit has its
conjugate orbit (with respect to ) belonging to or it is selfconjugate
then it is called selfconjugate. From lemma 1 we have that is the class of selfconjugate maps (with respect to ). Let us formulate this fact as theorem.
Theorem 4. The
class of selfconjugate maps is equal to.
Let us say
that involution contains involution writing if each transposition
of is also transposition
of . Let us clarify something about structure of selfconjugate
maps.
Theorem 5. (that is isomorphic to
normal subgroup of) is isomorphic to group .
Proof: Let , and as permutation acts on
universal set of corners , and is subdivision of that is induced by . In that case there exists an involution , that and in the same time orbits belong [as sets
of elements] either to or , i.e., if orbit belongs to , then belongs , or reversely. can be expressed as , where has corners belonging
to , and has corners belonging
to . But in that case, and are isomorphic to each
other and isomorphic to some permutation from and is isomorphic to some
permutation from , and is isomorphic to
permutation from . Theorem is proved.
How many
there are selfconjugate maps?
Theorem 6. .
Proof: ; .
How many
there are edge rotations, i.e., how many left coset has the group ?
Theorem 7. Group
has (itself including)
left cosets.
Proof: (including itself) has
as many left cosets as many edge rotations it is possible to generate, namely, . Indeed, there holds:
.