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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 458964, 8 pages
http://dx.doi.org/10.1155/2012/458964
Research Article

On the Periods of 2-Step General Fibonacci Sequences in the Generalized Quaternion Groups

Department of Mathematics, Science and Research Branch, Islamic Azad University, P.O. Box 14515-1775, Tehran 14778-93855, Iran

Received 12 October 2012; Accepted 3 December 2012

Academic Editor: Carlo Piccardi

Copyright © 2012 Bahram Ahmadi and Hossein Doostie. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study 2-step general Fibonacci sequences in the generalized quaternion groups . In cases where the sequences are proved to be simply periodic, we obtain the periods of 2-step general Fibonacci sequences.

1. Introduction

The study of the Fibonacci sequences in groups began with the earlier work of Wall [1] in 1960, where the ordinary Fibonacci sequences in cyclic groups were investigated. In the mid-eighties, Wilcox [2] extended the problem to the abelian groups. In 1990, Campbell et al. [3] expanded the theory to some classes of finite groups. In 1992, Knox proved that the periods of -nacci (-step Fibonacci) sequences in the dihedral groups are equal to , in [4]. In the progress of this study, the article [5] of Aydin and Smith proves that the lengths of the ordinary 2-step Fibonacci sequences are equal to the lengths of the ordinary 2-step Fibonacci recurrences in finite nilpotent groups of nilpotency class 4 and a prime exponent, in 1994.

Since 1994, the theory has been generalized and many authors had nice contributions in computations of recurrence sequences in groups and we may give here a brief of these attempts. In [6, 7] the definition of the Fibonacci sequence has been generalized to the ordinary 3-step Fibonacci sequences in finite nilpotent groups. Then in [8] it is proved that the period of 2-step general Fibonacci sequence is equal to the length of the fundamental period of the 2-step general recurrence constructed by two generating elements of a group of nilpotency class 2 and exponent . In [9] Karaduman and Yavuz showed that the periods of the 2-step Fibonacci recurrences in finite nilpotent groups of nilpotency class 5 and a prime exponent are , for , where is a prime and is the period of the ordinary 2-step Fibonacci sequence. The main role of [10, 11] in generalizing the theory was to study the 2-step general Fibonacci sequences in finite nilpotent groups of nilpotency class 4 and exponent and to the 2-step Fibonacci sequences in finite nilpotent groups of nilpotency class and exponent , respectively.

One may consult [12, 13] to see the results of the Fibonacci sequences in the modular groups concerning the periodicity of 2-step Fibonacci sequences constructed by two generating elements.

Going on a detailed literature in this area of research, we have to mention certain essential computation on the Fibonacci lengths of new structures like the semidirect products, the direct products, and the automorphism groups of finite groups which have been studied in [1419]. Finally, we refer to [20] where Karaduman and Aydin studied the periodicity property of 2-step general Fibonacci sequences in dihedral groups and the goal of this paper is to calculate the periods of 2-step general Fibonacci sequences in the generalized quaternion groups.

Let be a finite group. A -nacci sequence in group is a sequence of group elements for which each element is defined by , This sequence of the group is denoted by . We also call a 2-nacci sequence of group elements a Fibonacci sequence of a finite group. A finite group is -nacci sequenceable if there exists a -nacci sequence of such that every element of the group appears in the sequence. 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. We denote the period of a -nacci sequence by . A sequence of group elements is called simply periodic with period if the first elements in the sequence form a repeating subsequence. For example, the sequence is simply periodic with period 6. The following theorem is well known.

Theorem 1.1 (see [4]). Every -nacci sequence in a finite group is simply periodic.

2. Main Theorems

The generalized quaternion group , is a group with a presentation of the form It is easy to see that is of order , has order , has order 4, and the relation holds for all .

First we consider the following lemma which will be used frequently without further reference.

Lemma 2.1. For an integer , where , and for , the following relations hold in :(i), where is any even integer;(ii), where is any odd integer.

Proof. We may use an induction method on for both parts, simultaneously.

Theorem 2.2. .

Proof. Obviously, the order of is 4. If , then and hence the sequence has period 6. If , then and so the period is 8. Now let . Then, the first elements of are Since is of order 4, this sequence reduces to Thus, It follows that for   . We also have It shows that .

Let be a finite nonabelian 2-generated group. A 2-step general Fibonacci sequence in the group is defined by , for , and the integers and . Now we study this sequence for group .

Theorem 2.3. Let and be integers. If () or (), then 2-step general Fibonacci sequence in is not simply periodic.

Proof. First we consider the case (mod ). Then the sequence is Obviously, the cycle does not begin again with and , and hence the sequence is not simply periodic.
Now let (mod 4). Four cases occur.
Case 1 ( (mod 4)). Then
Since the cycle does not begin again with and , the sequence is not simply periodic.
Case 2 ( (mod 4)). Then Clearly, the sequence is not simply periodic.
Case 3 ( (mod 4)). Note that is a central element of . Thus Similar to Case 1, the sequence is not simply periodic.
Case 4 ( (mod 4)). So Thus the sequence is not simply periodic.

Theorem 2.4. Let be an even integer.(i)If () and (), then 2-step general Fibonacci sequence in is simply periodic with period 2.(ii)If () and (), then 2-step general Fibonacci sequence in is simply periodic with period 3.(iii)If () and (), then 2-step general Fibonacci sequence in is simply periodic with period 6.

Proof. (i) Since is even, (mod 4) and (mod 4). Thus and the period is 2.
(ii) Since is even, (mod 4) and (mod 4). Note that and are odd integers. So the sequence reduces to and the period is 3.
(iii) Because is even, (mod 4) and (mod 4). Furthermore, and are odd integers. Then the sequence reduces to and the period is 6.

Theorem 2.5. Let be an odd integer.(i)If (), (), () and , then 2-step general Fibonacci sequence in is simply periodic with period .(ii)If (), (), () and , then 2-step general Fibonacci sequence in is simply periodic with period .

Proof. (i) By induction on , we can show that and . In particular, the period must be even. Now we have and so the period is .
(ii) By induction on , it may be shown that Therefore, the period must be even. Now we have and thus the period is .

Theorem 2.6. Let and be odd integers, () and .(i)If (), then 2-step general Fibonacci sequence in is simply periodic with period .(ii)If () and (), then 2-step general Fibonacci sequence in is simply periodic with period .(iii)If () and (), then 2-step general Fibonacci sequence in is simply periodic with period .

Proof. (i) Two cases occur.
Case ( (mod 4)). By induction on , we can prove that the following relations hold: Consequently, the period must be a multiple of 6. Now we have and hence the period is .
Case ( (mod 4)). The proof is similar to Case 1 except that
(ii) It is easily shown that the following relations hold for every : Particularly, the period must be a multiple of 3. Since is odd, then it is not a multiple of . Thus . Further, and hence the period is .
(iii) By induction on , we can prove the following relations: Therefore, the period must be a multiple of 3. Now we have and so the period is .

Acknowledgments

The authors would like to express their appreciations to the referees for the valuable and constructive comments regarding the presentation of this paper.

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