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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 $k(G)=k$”.

11B39Fibonacci and Lucas numbers, etc.
20D60Arithmetic and combinatorial problems on finite groups
20D15Nilpotent finite groups, $p$-groups