`Discrete Dynamics in Nature and SocietyVolume 2011, Article ID 639476, 13 pageshttp://dx.doi.org/10.1155/2011/639476`
Research Article

On the Basic k-nacci Sequences in Finite Groups

1Department of Mathematics, Faculty of Science and Letters, Kafkas University, 36100 Kars, Turkey
2Department of Mathematics, Faculty of Science, Atatürk University, 25240 Erzurum, Turkey

Received 13 December 2010; Accepted 16 March 2011

Abstract

We define the basic k-nacci sequences and the basic periods of these sequences in finite groups, then we obtain the basic periods of the basic k-nacci sequences and the periods of the k-nacci sequences in symmetric group , its subgroups, and binary polyhedral groups which related with these groups.

1. Introduction

The study of Fibonacci sequences in groups began with the earlier work of Wall [1], where the ordinary Fibonacci sequences in cyclic groups were investigated. In the mid-eighties, Wilcox extended the problem to Abelian groups [2]. The theory is expanded to some finite simple groups by Campbell et al. [3]. There, they defined the Fibonacci length of the Fibonacci orbit and the basic Fibonacci length of the basic Fibonacci orbit in a 2-generator group. The concept of Fibonacci length for more than two generators has also been considered; see, for example, [4, 5]. Also, the theory has been expanded to the nilpotent groups; see, for example, [6, 7]. Other works on Fibonacci length are discussed in, for example, [810]. Knox proved that the periods of k-nacci (k-step Fibonacci) sequences in dihedral groups were equal to [11]. Deveci, Karaduman, and Campbell examined the period of the k-nacci sequences in some finite binary polyhedral groups in [12]. Recently, k-nacci sequences have been investigated; see, for example, [13, 14].

This paper defines the basic k-nacci sequences and the periods of these sequences in finite groups and discusses the basic periods of the basic k-nacci sequences and the periods of the k-nacci sequences in the symmetric group , alternating group , four-group, and binary polyhedral groups and with related and , respectively. We consider the groups , , , and both as 2-generator and as 3-generator groups.

A k-nacci sequence in a finite group is a sequence of group elements for which, given an initial (seed) set , each element is defined by We also require that the initial elements of the sequence generate the group, thus forcing the k-nacci sequence to reflect the structure of the group. The k-nacci sequence of a group generated by is denoted by [11].

A sequence of group elements is periodic if, after a certain point, it consists only of repetitions of a fixed subsequence. The number of elements in the repeating subsequence is called the period of the sequence. For example, the sequence is periodic after the initial element and has period 4. A sequence of group elements is simply periodic with period k if the first k elements in the sequence form a repeating subsequence. For example, the sequence is simply periodic with period 6. In [11], Knox had denoted the period of a k-nacci sequence by .

Definition 1.1. For a finitely generated group , where , the sequence , , , is called the Fibonacci orbit of with respect to the generating set , denoted as [4].

Definition 1.2. If is simply periodic, then the period of the sequence is called the Fibonacci length of with respect to generating set , written, [4].
Notice that the orbit of a k-generated group is a k-nacci sequence.
Let be a finite -generator group, and let be the subset of such that if and only if is generated by . We call a generating -tuple for .

2. Basic Period of Basic k-nacci Sequence

To examine the concept more fully, we study the action of automorphism group Aut of on and on the k-nacci sequences , . Now, Aut consists of all isomorphism and if Aut and , then .

For a subset and Aut, the image of under is

Definition 2.1. For a generating pair , the basic Fibonacci orbit of the basic length is defined by the sequence of elements of such that where is the least integer with for some Aut. Since generate , it follows that is uniquely determined. For more information, see [3].

Lemma 2.2. Let and let Aut , then .

Proof. Let . The result is obvious since and Each generating -tuple maps to distinct elements of under the action of elements of Aut. Hence, there are (where means the number of elements of ) nonisomorphic generating -tuples for . The notation was introduced in [15].
Suppose that elements of Aut map into itself, then there are distinct k-nacci sequences for Aut.

Definition 2.3. For a -tuple , the basic k-nacci sequence of the basic period is a sequence of group elements for which, given an initial (seed) set , each element is defined by where is the least integer with for some Aut. Since is a finite -generator group and , , , generate , it follows that is uniquely determined. The basic k-nacci sequence is finite containing element.
In this paper, we denote the basic period of the basic k-nacci sequence by .
From the definitions, it is clear that the periods of the k-nacci sequences and the basic k-nacci sequences in a finite group depend on the chosen generating set and the order of the generating elements.

Theorem 2.4. Let be a finite group and . If and , then divides , and there are elements of Aut which map into itself.

Proof. We have where is the order of automorphism Aut since and . Clearly, map into itself.

3. Applications

Definition 3.1. The polyhedral group for is defined by the presentation or The polyhedral group is finite if and only if the number is positive, that is, in the cases , , , and . Its order is . , , and are the groups , , and , respectively. Also, the groups , , and being isomorphic to the groups of rotations of the regular tetrahedron, octahedron, and icosahedron. Using Tietze transformations, we may show that . For more information on these groups, see [16] and [17, pp. 67-68].

Definition 3.2. The binary polyhedral group , for , is defined by the presentation or The binary polyhedral group is finite if and only if the number is positive. Its order is .

Definition 3.3. Let denote the th member of the -step Fibonacci sequence defined as with boundary conditions for and . Reducing this sequence by a modulo , we can get a repeating sequence, which we denote by where . We then have that , and it has the same recurrence relation as in (3.6) [18].

Theorem 3.4 ( is a periodic sequence [18]). Let denote the smallest period of , called the period of or the wall number of the k-step Fibonacci sequence modulo .

Theorem 3.5. The periods of the k-nacci sequences and the basic periods of the basic k-nacci sequences in the group are as follows.
if the group is defined by the presentation , then
(i)if and ,(ii)if and . If has the presentation , then(i′) if and ,(ii′) if , and .

Proof. Firstly, let us consider the 3-generator case. We first note that , , and (where means the order of ).
(i)If , we have the sequence for the generating triple (, , ), which has period 18 and the basic period 9 since , , and , where is the inner automorphism induced by conjugation by . (ii)If , we have the sequence for the generating triple (, , ), which has period 24 and the basic period 12 since , , and where is an outer automorphism of order 2.
If , the first elements of sequence for the generating triple (, , ) are Thus, using the above information, sequence reduces to where for . Thus, where for , , , , and .
We also have Since the elements succeeding , , and depend on , , and for their values, the cycle begins again with the element, that is, . Thus, .
It is easy to see from the above sequence that since , , and where is an outer automorphism of order 2.
Secondly, let us consider the 2-generator case. We first note that , , and .
(i′) If and since and where is the inner automorphism induced by conjugation by .(ii′)If , and since and where is an outer automorphism of order 2. The proofs are similar to above and are omitted.

Theorem 3.6. The periods of the k-nacci sequences and the basic periods of the basic k-nacci sequences in the binary polyhedral group are as follows.
If the group is defined by the presentation , then
(i)if and ,(ii)if and . If the group is defined by the presentation , then(i′) if and ,(ii′) if , and .

Proof. Firstly, let us consider the 2-generator case. We first note that , , and .
(i′) If , we have the sequence for the generating pair (, ), which has period 18 and the basic period 9 since and where is a outer automorphism of order 2. (ii′) If , we have the sequence for the generating pair (, ), which has period 24 and the basic period 24 since and where is an inner automorphism induced by conjugation by .
If , we have the sequence for the generating pair (, ), which has period 30 and the basic period 30 since and where is an inner automorphism induced by conjugation by .
If , the first elements of sequence for the generating pair (, ) are Thus, using the above information, sequence reduces to where . Thus, where , and denote the wall number of the -step Fibonacci sequence modulo 6.
If the group is defined by the presentation , then
if , and , if , where .

Proof. Firstly, let us consider the 3-generator case. We first note that , , and .(i)If and since , , and where is an outer automorphism of order 4. (ii)If let , then and since , , and where is an inner automorphism induced by conjugation by ;
let , then and since , , and where is an inner automorphism induced by conjugation by .
The proofs are similar to the proofs of Theorems 3.5.(i) and 3.5.(ii) and are omitted.
Secondly, let us consider the 2-generator case. We first note that , , and .
(i′) If and since and where is an outer automorphism of order 4. (ii′) If , let , then and since , , and where is an inner automorphism induced by conjugation by ,
let , then and since and where is an inner automorphism induced by conjugation by .
The proofs are similar to the proofs of Theorem 3.6.(i′) and Theorem 3.6.(ii′) and are omitted.

Theorem 3.9. The periods of the k-nacci sequences are , and the basic period of the basic k-nacci sequences is in   four-group.

Proof. We have the presentation . ; see [14] for a proof and since and where is an inner automorphism induced by conjugation by .

Acknowledgments

The authors thank the referees for their valuable suggestions which improved the presentation of the paper. This paper was supported by the Commission for the Scientific Research Projects of Kafkas Univertsity, Project no. 2010-FEF-61.

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