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

DOI: 10.1155/2012/458964

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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 [14–19]. Finally, we refer to [20] where Karaduman and Aydin studied the

References

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