Comments On The Fibonacci Sequences In Finite Groups

Transcription

1 Comments On The Fibonacci Sequences In Finite Groups Ömür DEVECĐ & Erdal KARADUMAN Atatürk University Department of Mathematics Faculty of Sicences 0 Erzurum / TURKEY odeveci6@hotmailcom eduman@atauniedutr Abstract In the work of Kno Steven W (99) the claim is made that A k-nacci sequence in a finite group is simply periodic [] We provide an eample to demonstrate that the claim is false Introduction The study of Fibonacci sequences in groups began with the earlier work of Wall [] where the ordinary Fibonacci sequences in cyclic groups were investigated In the mid eighties Wilco etended the problem to abelian groups [] Prolific co-operation of Campbell Doostie and Robertson epanded the theory to some finite simple groups [] Aydın and Smith proved in [] that the lengths of ordinary -step Fibonacci sequences are equal to the lengths of ordinary -step Fibonacci recurrences in finite nilpotent groups of nilpotency class and a prime eponent The theory has been generalized in [78] to the ordinary -step Fibonacci sequences in finite nilpotent groups Then it is shown in [] that the period of - step general Fibonacci sequence is equal to the length of fundamental period of the -step general Fibonacci sequence constructed by two generating elements of the group of eponent p and nilpotency class Karaduman and Yavuz showed that the periods of the -step Fibonacci recurrences in finite nilpotent groups of nilpotency class and a prime eponent are p k( p ) for < p 97 where p is prime and k( p ) is the periods of ordinary -step Fibonacci sequences[0] Kno proved that periods of k-nacci (k-step Fibonacci) sequences in

2 dihedral group were equal to k+ [] Recently the works have been done on Fibonacci sequences See for eample [69] A k-nacci sequence in a finite group is a sequence of group elements 0 L n L for which given an initial (seed) set 0 L j each element is defined by n 0L n for j n< k = n k n k+ L n for n k We also require that the initial elements of the sequence 0 L j 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 0 L j is denoted by Fk ( G; 0 L j ) and its period is denoted by Pk ( G; 0 L j ) Definition A -step Fibonacci sequence of a group elements is called a Fibonacci sequence of a finite group Definition A finite group G is k-nacci sequenceable if there eists a k-nacci sequence of G such that every element of the group appears in the sequence Definition A sequence of group elements is periodic if after a certain point it consists only of repetitions of a fied subsequence The number of elements in the repeating subsequence is called period of the sequence For eample the sequence a b c d e b c d e b c d e L is periodic after the initial element a and has period Definition A sequence of group elements is simply periodic with period k if the first k elements in the sequence form a repeating subsequence For eample the sequence a b c d e f g a b c d e f g a b c d e f g L is simply periodic with period 7 The following appears in [ Theorem ] Theorem : A k-nacci sequence in a finite group is simply periodic []

3 Definition The binary polyhedral group < l m n> for l m n> is defined by the presentation l m n < y z : = y = z = yz> Now we will give an eample satisfying the condition of [ Theorem ] but the -nacci sequence in a finite group is not to be necessary simply periodic Eample 7 Let us consider the binary polyhedral group < n> for n > defined by the presentation n < y z : = y = z = yz> The order the group defined by this presentation is n and the order of z is n and the orders of and y are Thus from relations in the group we have = yz z z = y = y Using definition of a k-nacci sequence in a finite group we obtain -nacci sequence in the group defined by this presentation as follow: = 8 9 M = y = z = yz= = z= y = y = yy = yy = yy y = = y 6 = yz= = = z= = z =

4 It is clear that -nacci sequence in the group < n> is periodic with period 6 but not simply periodic REFERENCES [] Aydın H & Dikici R " General Fibonacci Sequences in Finite Groups " Fibonacci Quarterly 6 (998) 6- [] Aydın H& Smith G C " Finite p -quotients of Some Cyclically Presented Groups" J London Math Soc 9 (99) 8-9 [] Campbell C M & Campbell PP The Fibonacci Length of Binary Polyhedral Groups and Related Groups Applications of Fibonacci Numbers 0 Kluwer (Dordrecht 006) 8-9 [] Campbell C M & Campbell PP The Fibonacci Length of Certain Centro- Polyhedral Groups J Appl Math &Computing 9 No-(00) -0 [] Campbell C M Doostie H & Robertson E F " Fibonacci Length of Generating Pairs in Groups" Applications of Fibonacci Numbers :7- Ed G E Bergum et al Kluwer Academic Publishers (990) [6] Campbell C M & Campbell PP Doostie H & Robertson E F " Fibonacci Lengths for Certain Metacyclic Groups Algebra Colloquium : (00) - [7] Dikici R & Smith G C " Recurrences in Finite Groups" Turkish J Math 9 (99) -9 [8] Dikici R & Smith G C " Fibonacci Sequences in Finite Nilpotent Groups" Turkish J Math (997)- [9] Doostie H & Hashemi M " Fibonacci Lengths Involving the Wall Number k( n ) J Appl Math Comput 0 No - (006) 7-80 [0] Karaduman E & Yavuz U On The Period of Fibonacci Sequences in Nilpotent Groups Applied Mathematics and Computation (00) -

5 [] Kno Steven W "Fibonacci Sequences in Finite Groups" Fibonacci Quarterly 0 (99) 6-0 [] Wall D D " Fibonacci Series Module m " Amer Math Monthly 67 (969) - [] Wilco H J " Fibonacci Sequences of Period n in Groups Fibonacci Quarterly (986) 6-6 Keywords: Fibonacci sequences Period Binary polyhedral group Mathematics Subject Classification Number : 0F0 0D60 B9