Classes of combinatorial maps with fixed edge rotation.

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:

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Let us define class of combinatorial maps that contain maps with fixed edge rotation:

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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:

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 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:

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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

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We used the fact that there holds . Theorem is proved.

 

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:

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