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Recurrences in finite groups. (English) Zbl 0866.11011
In a finite p-group G, let (r i ) be a 3-step Fibonacci sequence defined by the recurrence r i+3 =r i +r i+1 +r i+2 with given initial terms r 0 ,r 1 ,r 2 . It is trivial that the sequence (r i ) is periodic. Let k(G) be the least common multiple of the fundamental periods of all sequences in G satisfying the recurrence and denote by k the fundamental period of the sequence when r 0 =r 1 =0, r 2 =1. The main theorem of the paper is the following: “Let p>3 be a prime number, then if G is a non-trivial finite p-group of exponent p and nilpotency class 2, then k(G)=k”.
MSC:
11B39Fibonacci and Lucas numbers, etc.
20D60Arithmetic and combinatorial problems on finite groups
20D15Nilpotent finite groups, p-groups