Properties of Suborbits of the Dihedral Group Dn Acting on Ordered Subsets

A group action on a set is a process of developing an algebraic structure through a relation defined by the permutations in the group and the elements of the set. The process suppresses most of the group properties, emphasizing the permutation aspect, so that the algebraic structure has a wider application among other algebras. Such structures not only reveal connections between different areas in Mathematics but also make use of results in one area to suggest conjectures and also prove results in a related area. The structure (G, X) is a transitive permutation group G acting on the set X. Investigations on the properties associated with various groups acting on various sets have formed a subject of recent study. A lot of investigations have been done on the action of the symmetric group Sn on various sets, with regard to rank, suborbits and subdegrees. However, the action of the dihedral group has not been thoroughly worked on. This study aims at investigating the properties of suborbits of the dihedral group Dn acting on ordered subsets of { } 1,2, , X n =  . The action of Dn on X[r], the set of all ordered r-element subsets of X, has been shown to be transitive if and only if n = 3. The number of self-paired suborbits of Dn acting on X[r] has been determined, amongst other properties. Some of the results have been used to determine graphical properties of associated suborbital graphs, which also reflect some group theoretic properties. It has also been proved that when G = Dn acts on ordered adjacent vertices of G, the number of self-paired suborbits is n + 1 if n is odd and n + 2 if n is even. The study has also revealed a conjecture that gives a formula for computing the self-paired suborbits of the action of Dn on its ordered adjacent vertices. Properties of suborbits are significant as they form a link between group theory and graph theory.


Introduction
Previous investigations on rank, suborbits and subdegrees have taken into account the symmetric group S n acting on various subsets of { } 1, 2, , X n =  .The rank and subdegrees of S n on 2-element subsets were shown to be 3 and 1, 2, respectively [1].The study was generalized to the action of S n on X (r) , unordered r-element subsets of where it was established that the rank is r + 1 if n ≥ 2r, and the sudegrees are; 1, n r r n r n r r r r It was also proved that the suborbits of S n acting on X (r) are self-paired [2].Similarly, the action of S n on ordered r-element subsets of was discussed.The action was shown to be transitive, the rank of S n on X [3] as 34 for n ≥ 6, and the rank of S n on X [2] as 7 for n ≥ 4. Properties of suborbital graphs were also examined in this action [3].The action of the dihedral group D n on X (r) has also been considered where the action was proved transitive for values of r = 1 and r = n − 1 [4].
The study of suborbits of the dihedral group acting on ordered subsets has revealed some interesting properties which translate to properties of associated suborbital graphs.This has seen a clear connection between Group Theory and Graph Theory, which has realized the artistic value in Mathematics.Section 2 outlines some of the notations and preliminary definitions which have been used in the investigations.Section 3 discusses the rank, subdegrees and suborbits of D n on X [r] .Some properties of suborbits have also been discussed in this Section.
Section 4 has seen the application of properties of suborbits in construction of associated suborbital graphs.Section 5 examines suborbits of D n on its ordered adjacent vertices.
The dihedral group D n consists of all the symmetries of a regular n-sided polygon.The group is of order 2n, constituted by n rotations and n reflections.

Notations and Preliminary Definitions
Notation 2.1 Throughout this paper, G is the dihedral group D n , X [r] is the set of all ordered r-element subsets of , and n P r is n permutation r.
Definition 2.2 (Group action) [5] Let X be a non-empty set.The group action of G on X is a relation on the pair (G, X) such that gx is a unique image of every x in X under g in G.The relation satisfies the algebraic laws of identity and associativity.Namely, , for all g 1 , g 2 in G and x in X Definition 2.3 (Orbit of an element) (see [5]: p. 31) A group action partitions the set into disjoint equivalence classes known as G-orbits.The orbit of each x in X is the subset of X, which contains all the images of x under every g in G. Let G be a group acting transitively on a set X.The stabilizer of x in X is the set of all elements g in G such that gx = x.The set is denoted by

{ }
, for a fixed i : . The G x -orbits on X, 0 1 1 , , , m− ∆ ∆ ∆  are known as suborbits of G.The rank of G is m and the cardinalities, ( ) , are the subdegrees of G.It was established in [6] that both m and the cardinalities of the suborbits are independent of the choice of x in X.

Definition 2.5
Let G act transitively on a set X and let ∆ be an orbit of G x on X. Define Theorem 2.6 (see [8]) Let G act transitively on a set X, and suppose g ∈ G.The number of selfpaired suborbits of G is given by ( ) Theorem 2.7 (see [9]) Let G act transitively on X and let G x be the stabilizer of the point x X ∈ .Let , 0,1, , Let G be a group acting on a finite set X with x in X.The size of the orbit of x in G is the index ( ) Theorem 2.9 (Cauchy-Frobenius Lemma [11]: p. 98) Suppose G is a group acting on a finite set X.The number of G-orbits on X is given by ( ) where ( ) Fix g denotes the number of elements in X fixed by g in G.

Rank, Subdegrees and Suborbits of G = Dn on X [r]
The action of G on X [r] is defined by the rule; [ ] [ ] , , , , , , , , , r x x x  in X [r] .The set X [r] consists of all permutations of r elements from , and its cardinality is n P r .
Theorem 3.1.1 The action of G = D n on X [r] is transitive if and only if n = 3. Proof: , , , r x x x  in X [r] .Now, g fixes an ordered r-element , , , , , , . This is possible only if g is the identity.It follows that the number of elements in X [r] fixed by g is n P r .Using Theorem 2.9, the number of G-orbits on X [r] is; The least number of G-orbits on X [r] is realized when n = r, where the number is ( ) . If the action is transitive, then ( ) = , and the action is transitive.
Example 3.1.2 The set X [2] consists of all ordered 2-element subsets of Hence, its cardinality is n P 2 = ( ) Let [x, y] be in X [2] .Now, g in G fixes an ordered pair, [x, y], if and only if g x y x y = , so that gx = x and gy = y.When n is odd, this is possible only if g is the identity.It follows that, . Using Theorem 2.9, the number of G-orbits on X [2] is ( ) . Since n = 3, the number of G-orbits is 1 and G = D 3 acts transitively on X [2] .
When n is even, g fixes an ordered pair if g is the identity or a reflection through a diagonal.If g is the identity, ( ) Fix g is (n P 2 ).Since each reflection fixes 2 elements of X [2] and the number of such reflections is n/2, then ( ) The number of G-orbits on X [2] is; ( ) For transitivity, But, n ≥ 3 for G to be defined.It follows that the action is intransitive.The rank of G = D 3 on X [r] is 6 and each suborbit contains 1 element.

Proof:
Let the group G = D 3 act on X [r] and [ The rank of G = D 3 on X [3] is 6 and each suborbit is of length 1.

Some Properties of suborbits of D3 on X [r]
Some properties of suborbits of D 3 acting on X [r] have been discussed in the following Theorems.
Theorem 3.2.1 The number of self-paired suborbits of D 3 on X [r] is 4. Proof: The number of self-paired suborbits is determined by the fixed point set of g 2 , by Theorem 2.6.
If r = 2, then g 2 fixes an ordered set of 2 elements if g is the identity or g is a reflection.If g is the identity, then ( ) If g is a reflection, then ( ) Since the number of reflections is 3, then the number of self-paired suborbits is; . The 4 self-paired suborbits of D 3 on X [2] are; , , g ∆ ∆ = ∆ ∆ when g = (13).If r = 3, then g 2 fixes an ordered set of 3 elements if g is the identity or a reflection.Computation of self-paired suborbits is similar to the case when r = 2.The 4 self-paired suborbits of D 3 on X [3] are; . Clearly, g can be chosen accordingly so that; [ ] [ ] , , , X [r] , where Proof: . This implies that g(1) = x 1 and g(x 1 ) = 1, ⇒ ( ) . Hence, g 2 = 1.Conversely, if g 2 = 1, then g , , , 1,2, , Suppose G = D 3 is transitive on X [r] .Then ∆ i and ∆ j are paired suborbits of G if and only if the permutations g and h, in the maps g∆ i = ∆ 0 and h∆ j = ∆ 0 , are inverses of each other. Proof: Suppose ∆ i and ∆ j are paired suborbits of G. Then there exists g and h in G such that g∆ i = ∆ 0 and h∆ j = ∆ 0 .There exists x i in ∆ i and y i in ∆ j such that gx i = 1 and hy i = 1.By Definition 2.5, g(1) = y i and h(1) = x i .It follows that, gh(1) = 1 and hg(1) = 1.Hence, g and h are inverses of each other.Conversely, if g and h are inverses of each other, then g maps x i to 1 and 1 to y i .Similarly, h maps y i to 1 and 1 to x i .Hence, ∆ i and ∆ j are paired suborbits.

Suborbital Graphs of G
be a suborbit of G on X [r] , where { } . The graph ᴦ corresponding to suborbital O is formed by considering X [r] as the vertex set and drawing an edge from [ ] , , , , , , , All suborbital graphs corresponding to the action of D 3 on X [r] are disconnected. Proof: Let ᴦ i be the suborbital graph corresponding to a self-paired suborbit ∆ i .Sup- , , , r c c c  is a point on the vertex set, X [r] .Then there is an edge from [ ] The action of G = D 3 on X [r] is imprimitive.

Proof:
Suppose ᴦ i is a suborbital graph corresponding to a suborbit, ∆ i , of G on X [3] .

 6 rX
acts by fixing each element of X[r] in its own[ ] 1,2, ,r G  -orbit.Since [ ] =then the rank of G on X[r] is 6.Clearly, the subdegrees of G are; 1, 1, 1, 1, 1, 1. Advances in Pure Mathematics Example 3.1.4 coordinates, c i and d i , are identical.The 3 coordinates can be rearranged in 3! ways.Each coordinate can take the same position in 1/3(3!) = 2 possibilities.Thus, an edge joining exactly 2 vertices is a connected component and the graph is disconnected.If ∆ i is a paired suborbit, then the corresponding graph has an edge from the corresponding coordinates, c i and d i , are identical.The number of such rearrangements is 1/2(3!) = 3.It follows that a path joining exactly 3 vertices is a connected component.Hence, the graph is disconnected.Theorem 4.2