Combinatorial Properties of the Alternating & Dihedral Groups and Homomorphic Images of Fibonacci Groups

Combinatorial Properties of the Alternating & Dihedral Groups and Homomorphic Images of Fibonacci Groups

Chapter One

OBJECTIVE OF THE STUDY

The objective of this research is to

1. Obtain the number of even (odd) permutations having exactly kfixed points in the alternating group, discuss the fixed points and the generating functions for the fixed points.
2. Give two different proofs one geometric and the other algebraic (in line with Catarino and Higgins 1999) of the number of even and odd permutation (of an n – element set) having exactly k fixed points in the dihedral group. In the algebraic proof, we further obtain the formulae for determining the fixed points.
3. Prove the three families; F (2r,4r + 2), F (4r + 3, 8r + 8) and F (4r + 5,8r + 12) of the Fibonacci groups F (m , n) to be infinite by defining Morphism between Dihedral groups and the Fibonacci groups.
4. Obtain the number of permutations of Xn  that can be expressed as a product of r_i (m – i +1, i = 1, 2, ⋯, m -1) cycles.

CHAPTER TWO

LITERATURE REVIEW

TRANSITIVE PERMUTATION GROUPS

Let G be a permutation group on W and D a subset of W , D is said to be a fixed block of G if

D^G =D or D ÇD^G ¹ ϕ .

The union and intersection of any two fixed blocks is a fixed block. Every group G in W has two trivial fixed blocks ϕ and W

Orbit 0f  α in G The fixed block

D ¹ ϕ

is called an orbit or set of transitivity of G on W, denoted by α ^G or

αG where αG is defined as α ^G := {α ^g

g Î G}, α Î W

A group G acting on a set W is said to be a transitive permutation group if it has only one orbit i.e. α ^G = W .Thus, for all

α , β ÎW

there exists g ÎG such that

α ^g = β .

A group which is not transitive is called intransitive. A group G acting transitively on a set W is said to act regularly if

α ^G = 1 for each α ÎW, that is only the identity fixes any point. The number of elements in α ^G is called the length of the orbit.

A relation ~ in W defined by the rule, α ~ β Þ α ^g = β

” g Î G, α , β ÎW with αg = β is an equivalence relation.

The orbits of G partition W , for let

D1 , D 2 , D 3 , …, D S be the orbits o G on W then G induces a permutation group G ^W on D and D is a disjoint union of orbits

 D = ∪DI i =1

. Moreover, G £ ÕG ^DI i =1

and we say that G is a direct product of the groups

G^D1 , G^D2 , … , G^DS . If also, each

GD I (i = 1, 2 ,…, s.) is isomorphic to a group H (possibly H £ G ). We say that G is a sub-direct product of H .

A subset D of W is said to be G – invariant if for all

g Î G, β Î D and β ^g = s implies β ^g Î D

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CHAPTER THREE

RESULTS

 RESULT ONE

SOME COMBINATORIAL PROPERTIES OF THE ALTERNATING

GROUP

Let Xn = {1, 2, …, n} be a finite n -element set and let Sn , In , and An be as defined, the combinatorial properties of Sn have been studied over long period and many interesting results have emerged. In particular, the number of permutations of ( X n )

having exactly k fixed points and their generating functions are known.

In this section we obtain and discuss formulae for the number of even permutations (of an n -element set) having exactly k fixed points. Moreover, we obtain generating functions for these numbers. We also obtain similar results for the number of odd permutations.

We list some combinatorial results, (some may be found in chapter two and one), that we shall need later in our proofs.

CHAPTER FOUR

SUMMARY OF RESULTS, CONTRIBUTIONS AND AREAS FOR FURTHER RESEACH

SUMMARY OF RESULTS

We have, in this thesis, accomplished the following:

1. We obtained and discussed formulae for the number of even permutations (of ann -element set) having exactly k fixed points in the alternating group.
2. We obtained generating functions for the number of even permutations having exactlyk fixed points in alternating group.
3. We also obtained similar results (as in 1 and 2 above) for the number of odd permutations having exactly k fixed points and their generating functions in the alternating group.
4. We give a geometric proof for the number of even (odd) permutations (of an n -element set) having exactly k fixed points in the dihedral group.
5. We give an algebraic proof in line of Catarino and Higgins (1999) for the number of even (odd) permutations having exactly k  fixed points, in the dihedral group.
6. We proved the three families: F (2r,4r + 2), F (4r + 3,8r + 8 and F (4r + 5,8r + 12) of the Fibonacci groups F (m , n) to be infinite by defining morphism between Dihedral groups and the Fibonacci groups.
7. We give an alternative prove of the Cauchy’s formula f (m , n) for be thenumber of permutations of Xn that can be express as a product  of r_i (m – i +1, i = 1, 2, ⋯, m -1) cycles.

 CONTRIBUTIONS TO KNOWLEDGE

2. We obtained and discussed formulae for the number of even and odd permutations (of an n – element set) having exactly k fixed points in the alternating group and the generating functions for the fixed points.
3. We give two different proofs of the number of even and odd permutations (of an n – element set) having exactly k fixed points in the dihedral group, on geometric and the other algebraic. In the algebraic proof, however, we further obtain the formulae for determining the fixed points.
4. We proved the three families; F (2r,4r + 2), F (4r + 3,8r + 8) and F (4r + 5,8r + 12) of the Fibonacci groups F (m , n) to be infinite by defining Morphism between Dihedral groups and the Fibonacci groups.
5. We give an alternative prove of the Cauchy’s formula for the number permutations with a given cycle structure.

AREAS FOR FURTHER RESEACH

• Thenew method we introduced may be tested for the two families F (7 + 5i, 5) and F (8 + 5i, 5) for integers B ³ 0that remain unsettled by creating morphism between the Fibonacci groups and a suitable permutation group
• There is room for further research in the determination of more combinatorial properties of the permutation groups we discussed and other permutation groups.
• The study of classification of transitive p groups of degree say p^m in line with Audu (1986a) can be considered, by obtaining the number of k  fixed points and the generating functions for the fixed points of transitive p groups of degree say p^m .
• The study of permutations as even(odd) according to its length can be considered using number of fixed points.
• The number of even (odd) permutations with a given cycle
• The number of cycle structures in a given

REFERENCES

• Apine, E. (2000) On Transitive p-Groups of Degree p² or p³. PhD Thesis University of Jos.
• Audu, M. S. (1986a). Generating sets for transitive permutation groups of prime-power order. Abacus 17 (2): 22-26.
• Audu, M. S. (1988b). The structure of the permutation modules for transitive p-groups of degree p². Journal of Algebra 17:227-239.
• Audu, M. S. (1988c). The structure of the permutation modules for transitive abelian groups of prime-power order. Nigerian Journal of Mathematics and Applications 17: 1-8.
• Audu, M. S. (1988d). The number of transitive p-groups of degree p². Advances in Modeling and Simulation Enterprises Review 7(4): 9-13.
• Audu, M. S. (1989e). Groups of prime-power order acting on models over a modular field. Advances in Modeling and Simulation Enterprises Review 9 (4): 1-10.
• Audu, M. S. (1991f). On transitive permutation groups. Afrikan Mathematika Journal of African Mathematical Union 4(2): 155-60.
• Audu, M. S., Momoh, S. U. & Apine, E. (1994). On the classification of transitive p-groups of degree p³. Nigerian Journal of Mathematics and Applications 7:1-12.
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• Catarino, P.M. (2000). Monoids of orientation-preserving transformations of a finite chain and their presentation. Semigroup Forum 60:262-276.
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• Cemeron, P. J. (2000). Sequences realized by oligomorphic permutation groups. Journal. Integer Sequence 30:1.5.
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