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

Abstract

We study 2-step general Fibonacci sequences in the generalized quaternion groups 𝑄_4𝑛. 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 𝑘-nacci (𝑘-step Fibonacci) sequences in the dihedral groups are equal to 2𝑘+2, 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 𝑝. 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 𝑝⋅𝑘(𝑝), for 2<𝑝≤2927, where 𝑝 is a prime and 𝑘(𝑝) 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 𝑝 and to the 2-step Fibonacci sequences in finite nilpotent groups of nilpotency class 𝑛 and exponent 𝑝, 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 [14–19]. 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 𝐺=⟨𝑎₀,𝑎₁,𝑎₂,…,𝑎_𝑗−1⟩ be a finite group. A 𝑘-nacci sequence in group 𝐺 is a sequence {𝑥_𝑛}^∞_𝑛=0 of group elements for which each element is defined by 𝑥₀=𝑎₀,𝑥₁=𝑎₁,…,𝑥_𝑗−1=𝑎_𝑗−1, 𝑥_𝑛=𝑥₀𝑥₁⋯𝑥_𝑛−1𝑥,for𝑗≤𝑛<𝑘,_𝑛−𝑘𝑥_{𝑛−𝑘+1}⋯𝑥_𝑛−1,for𝑛≥𝑘. (1.1) This sequence of the group 𝐺 is denoted by 𝐹_𝑘(𝐺;𝑎₀,𝑎₁,𝑎₂,…,𝑎_𝑗−1). We also call a 2-nacci sequence of group elements a Fibonacci sequence of a finite group. A finite group 𝐺 is 𝑘-nacci sequenceable if there exists a 𝑘-nacci sequence of 𝐺 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 𝑎,𝑏,𝑐,𝑑,𝑑,𝑒,𝑏,𝑐,𝑑,𝑒,… is periodic after the initial element 𝑎 and has period 4. We denote the period of a 𝑘-nacci sequence 𝐹_𝑘(𝐺;𝑎₀,𝑎₁,𝑎₂,…,𝑎_𝑗−1) by 𝑃_𝑘(𝐺;𝑎₀,𝑎₁,𝑎₂,…,𝑎_𝑗−1). A sequence of group elements is called simply periodic with period 𝑘 if the first 𝑘 elements in the sequence form a repeating subsequence. For example, the sequence 𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,… is simply periodic with period 6. The following theorem is well known.

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

2. Main Theorems

The generalized quaternion group 𝑄_4𝑛, (𝑛≥2) is a group with a presentation of the form 𝑎,𝑏∣𝑎^2𝑛=1,𝑎^𝑛=𝑏²,𝑏⁻¹𝑎𝑏=𝑎⁻¹. (2.1) It is easy to see that 𝑄_4𝑛 is of order 4𝑛, 𝑎 has order 2𝑛, 𝑏 has order 4, and the relation 𝑎^𝑘𝑏=𝑏𝑎^−𝑘 holds for all 𝑘∈ℤ.

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

Lemma 2.1. For an integer 𝑟, where 1≤𝑟<2𝑛, and for 𝑠∈{1,3}, the following relations hold in 𝑄_4𝑛:(i)(𝑎^𝑟𝑏^𝑠)^𝑡=𝑏^𝑠𝑡, where 𝑡 is any even integer;(ii)(𝑎^𝑟𝑏^𝑠)^𝑡=𝑎^𝑟𝑏^𝑠𝑡, where 𝑡 is any odd integer.

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

Theorem 2.2. 𝑃_𝑘(𝑄_4𝑛;𝑎,𝑏)=2𝑘+2.

Proof. Obviously, the order of 𝑎𝑏 is 4. If 𝑘=2, then 𝑥₀=𝑎,𝑥₁=𝑏,𝑥₂=𝑎𝑏,𝑥₃=𝑎^𝑛−1,𝑥₄=𝑎^𝑛+2𝑏,𝑥₅=𝑎𝑏,𝑥₆=𝑎,𝑥₇=𝑏,…, (2.2) and hence the sequence has period 6. If 𝑘=3, then 𝑥₀=𝑎,𝑥₁=𝑏,𝑥₂=𝑎𝑏,𝑥₃=(𝑎𝑏)²=𝑏²,𝑥₄=𝑎⁻¹,𝑥₅=𝑎²𝑏³,𝑥₆=𝑎𝑏,𝑥₇=1,𝑥₈=𝑎,𝑥₉=𝑏,…, (2.3) and so the period is 8. Now let 𝑘≥4. Then, the first 𝑘+1 elements of 𝐹_𝑘(𝑄_4𝑛;𝑎,𝑏) are 𝑥₀=𝑎,𝑥₁=𝑏,𝑥₂=𝑎𝑏,𝑥₃=(𝑎𝑏)²,𝑥₄=(𝑎𝑏)⁴,…,𝑥_𝑘=(𝑎𝑏)^2𝑘−2. (2.4) Since 𝑎𝑏 is of order 4, this sequence reduces to 𝑥₀=𝑎,𝑥₁=𝑏,𝑥₂=𝑎𝑏,𝑥₃=𝑏²,𝑥_𝑗=1,4≤𝑗≤𝑘. (2.5) Thus, 𝑥_𝑘+1=^𝑘_𝑖=1𝑥_𝑖=𝑏(𝑎𝑏)𝑏²=𝑎⁻¹,𝑥_𝑘+2=^𝑘+1_𝑖=2𝑥_𝑖=(𝑎𝑏)𝑏²𝑎⁻¹=𝑎²𝑏³,𝑥_𝑘+3=^𝑘+2_𝑖=3𝑥_𝑖=𝑏²𝑎⁻¹𝑎²𝑏³𝑥=𝑎𝑏,_𝑘+4=^𝑘+3_𝑖=4𝑥_𝑖=𝑎⁻¹𝑎²𝑏³(𝑎𝑏)=1. (2.6) It follows that 𝑥_𝑘+𝑗=1 for 𝑗  (4≤𝑗≤𝑘+1). We also have 𝑥_2𝑘+2=^2𝑘+1_𝑖=𝑘+2𝑥_𝑖=𝑎²𝑏³()𝑥𝑎𝑏=𝑎,_2𝑘+3=^2𝑘+2_𝑖=𝑘+3𝑥_𝑖=(𝑎𝑏)𝑎=𝑏. (2.7) It shows that 𝑃_𝑘(𝑄_4𝑛;𝑎,𝑏)=2𝑘+2.

Let 𝐺=⟨𝑎,𝑏⟩ be a finite nonabelian 2-generated group. A 2-step general Fibonacci sequence in the group 𝐺 is defined by 𝑥₀=𝑎,𝑥₁=𝑏,𝑥_𝑖=𝑥^𝑚_𝑖−2𝑥^𝑙_𝑖−1, for 𝑖≥2, and the integers 𝑚 and 𝑙. Now we study this sequence for group 𝑄_4𝑛.

Theorem 2.3. Let 𝑚 and 𝑙 be integers. If 𝑚≡0 (mod2𝑛) or 𝑚≡0 (mod4), then 2-step general Fibonacci sequence in 𝑄_4𝑛 is not simply periodic.

Proof. First we consider the case 𝑚≡0 (mod 2𝑛). Then the sequence is 𝑥₀=𝑎,𝑥₁=𝑏,𝑥₂=𝑎^𝑚𝑏^𝑙=𝑏^𝑙,𝑥₃=𝑏^𝑚𝑏^𝑙2=𝑏^𝑚+𝑙2,…. (2.8) Obviously, the cycle does not begin again with 𝑎 and 𝑏, and hence the sequence is not simply periodic.
Now let 𝑚≡0 (mod 4). Four cases occur.
Case 1 (𝑙≡0 (mod 4)). Then 𝑥₀=𝑎,𝑥₁=𝑏,𝑥₂=𝑎^𝑚,𝑥₃=𝑎^𝑚𝑙,𝑥₄=𝑎^{𝑚2+𝑚𝑙2},𝑥₅=𝑎^{2𝑚2𝑙+𝑚𝑙3},𝑥₆=𝑎^{𝑚3+3𝑚2𝑙2+𝑚𝑙4},…. (2.9)
Since the cycle does not begin again with 𝑎 and 𝑏, the sequence is not simply periodic.
Case 2 (𝑙≡1 (mod 4)). Then 𝑥₀=𝑎,𝑥₁=𝑏,𝑥₂=𝑎^𝑚𝑏,𝑥₃=𝑎^𝑚𝑏,𝑥₄=𝑎^𝑚𝑏,…. (2.10) Clearly, the sequence is not simply periodic.
Case 3 (𝑙≡2 (mod 4)). Note that 𝑏² is a central element of 𝑄_4𝑛. Thus 𝑥₀=𝑎,𝑥₁=𝑏,𝑥₂=𝑎^𝑚𝑏²,𝑥₃=𝑎^𝑚𝑙,𝑥₄=𝑎^{𝑚2+𝑚𝑙2},𝑥₅=𝑎^{2𝑚2𝑙+𝑚𝑙3},…. (2.11) Similar to Case 1, the sequence is not simply periodic.
Case 4 (𝑙≡3 (mod 4)). So 𝑥₀=𝑎,𝑥₁=𝑏,𝑥₂=𝑎^𝑚𝑏³,𝑥₃=𝑎^𝑚𝑏,𝑥₄=𝑎^𝑚𝑏³,𝑥₅=𝑎^𝑚𝑏,…. (2.12) Thus the sequence is not simply periodic.

Theorem 2.4. Let 𝑛 be an even integer.(i)If 𝑚≡1 (mod2𝑛) and 𝑙≡0 (mod2𝑛), then 2-step general Fibonacci sequence in 𝑄_4𝑛 is simply periodic with period 2.(ii)If 𝑚≡−1 (mod2𝑛) and 𝑙≡−1 (mod2𝑛), then 2-step general Fibonacci sequence in 𝑄_4𝑛 is simply periodic with period 3.(iii)If 𝑚≡1 (mod2𝑛) and 𝑙≡1 (mod2𝑛), then 2-step general Fibonacci sequence in 𝑄_4𝑛 is simply periodic with period 6.

Proof. (i) Since 𝑛 is even, 𝑚≡1 (mod 4) and 𝑙≡0 (mod 4). Thus 𝑥₀=𝑎,𝑥₁=𝑏,𝑥₂=𝑎,𝑥₃=𝑏,…, (2.13) and the period is 2.
(ii) Since 𝑛 is even, 𝑚≡3 (mod 4) and 𝑙≡3 (mod 4). Note that 𝑚 and 𝑙 are odd integers. So the sequence reduces to 𝑥₀=𝑎,𝑥₁=𝑏,𝑥₂=𝑎⁻¹𝑏³,𝑥₃=𝑎,𝑥₄=𝑏,…, (2.14) and the period is 3.
(iii) Because 𝑛 is even, 𝑚≡1 (mod 4) and 𝑙≡1 (mod 4). Furthermore, 𝑚 and 𝑙 are odd integers. Then the sequence reduces to 𝑥₀=𝑎,𝑥₁=𝑏,𝑥₂=𝑎𝑏,𝑥₃=𝑎^𝑛−1,𝑥₄=𝑎²𝑏³,𝑥₅=𝑎𝑏,𝑥₆=𝑎,𝑥₇=𝑏,…, (2.15) and the period is 6.

Theorem 2.5. Let 𝑛 be an odd integer.(i)If 𝑚≡1 (mod2𝑛), 𝑚≡1 (mod4), 𝑙≡0 (mod4) and (𝑙,𝑛)=1, then 2-step general Fibonacci sequence in 𝑄_4𝑛 is simply periodic with period 2𝑛.(ii)If 𝑚≡1 (mod2𝑛), 𝑚≡3 (mod4), 𝑙≡0 (mod4) and (𝑙,𝑛)=1, then 2-step general Fibonacci sequence in 𝑄_4𝑛 is simply periodic with period 4𝑛.

Proof. (i) By induction on 𝑘≥0, we can show that 𝑥_2𝑘=𝑎 and 𝑥_2𝑘+1=𝑎^−𝑘𝑙𝑏. In particular, the period must be even. Now we have 𝑥_2𝑘+1=𝑏⟺2𝑛∣𝑘𝑙⟺𝑛∣𝑘(because(𝑙,𝑛)=1), (2.16) and so the period is 2𝑛.
(ii) By induction on 𝑘≥0, it may be shown that 𝑥_2𝑘=𝑎,𝑥_2𝑘+1=𝑎^−𝑘𝑙𝑎𝑏if𝑘iseven,_−𝑘𝑙𝑏³if𝑘isodd. (2.17) Therefore, the period must be even. Now we have 𝑥_2𝑘+1=𝑏⟺2𝑛∣𝑘𝑙⟺2𝑛∣𝑘(because(𝑙,𝑛)=1and𝑛isodd), (2.18) and thus the period is 4𝑛.

Theorem 2.6. Let 𝑛 and 𝑙 be odd integers, 𝑚≡−1 (mod2𝑛) and (𝑙+1,𝑛)=1.(i)If 𝑚≡1 (mod4), then 2-step general Fibonacci sequence in 𝑄_4𝑛 is simply periodic with period 6𝑛.(ii)If 𝑚≡3 (mod4) and 𝑙≡1 (mod4), then 2-step general Fibonacci sequence in 𝑄_4𝑛 is simply periodic with period 6𝑛.(iii)If 𝑚≡3 (mod4) and 𝑙≡3 (mod4), then 2-step general Fibonacci sequence in 𝑄_4𝑛 is simply periodic with period 3𝑛.

Proof. (i) Two cases occur.
Case 1 (𝑙≡1 (mod 4)). By induction on 𝑘≥0, we can prove that the following relations hold: 𝑥_6𝑘𝑥=𝑎,_6𝑘+1=𝑎^{−2𝑘(𝑙+1)}𝑥𝑏,_6𝑘+2=𝑎^{−(2𝑘(𝑙+1)+1)}𝑥𝑏,_6𝑘+3=𝑎^𝑛+1,𝑥_6𝑘+4=𝑎^{−(𝑛+(2𝑘+1)(𝑙+1))}𝑥𝑏,_6𝑘+5=𝑎^{−((2𝑘+1)(𝑙+1)+1)}𝑏. (2.19) Consequently, the period must be a multiple of 6. Now we have 𝑥_6𝑘+1=𝑏⟺2𝑛∣2𝑘(𝑙+1)⟺𝑛∣𝑘(because(𝑙+1,𝑛)=1), (2.20) and hence the period is 6𝑛.
Case 2 (𝑙≡3 (mod 4)). The proof is similar to Case 1 except that 𝑥_6𝑘+2=𝑎^{−(𝑛+2𝑘(𝑙+1)+1)}𝑥𝑏,_6𝑘+4=𝑎^{−(2𝑘+1)(𝑙+1)}𝑏. (2.21)
(ii) It is easily shown that the following relations hold for every 𝑘≥0: 𝑥_3𝑘=𝑎,𝑥_6𝑘=𝑎,𝑥_6𝑘+1=𝑎^{−2𝑘(𝑙+1)}𝑏,𝑥_6𝑘+2=𝑎^{−(2𝑘(𝑙+1)+1)}𝑥𝑏,_6𝑘+3=𝑎,𝑥_6𝑘+4=𝑎^{−(𝑛+(2𝑘+1)(𝑙+1))}𝑏,𝑥_6𝑘+5=𝑎^{−(𝑛+1+(2𝑘+1)(𝑙+1))}𝑏. (2.22) Particularly, the period must be a multiple of 3. Since 𝑛+(2𝑘+1)(𝑙+1) is odd, then it is not a multiple of 2𝑛. Thus 𝑥_6𝑘+4≠𝑏. Further, 𝑥_6𝑘+1=𝑏⟺2𝑛∣2𝑘(𝑙+1)⟺𝑛∣𝑘(because(𝑙+1,𝑛)=1), (2.23) and hence the period is 6𝑛.
(iii) By induction on 𝑘≥0, we can prove the following relations: 𝑥_3𝑘𝑥=𝑎,_3𝑘+1=𝑎^{−𝑘(𝑙+1)}𝑥𝑏,_3𝑘+2=𝑎^{−(𝑛+1+𝑘(𝑙+1))}𝑏. (2.24) Therefore, the period must be a multiple of 3. Now we have 𝑥_3𝑘+1=𝑏⟺2𝑛∣𝑘(𝑙+1)⟺𝑛∣𝑘(because(𝑙+1,𝑛)=1), (2.25) and so the period is 3𝑛.

Acknowledgments

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