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Zbl 0866.11011
Dikici, R.; Smith, G.C.
Recurrences in finite groups.
(English)
[J] Turk. J. Math. 19, No.3, 321-329 (1995). ISSN 1300-0098; ISSN 1303-6149/e

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\$''.
[Péter Kiss (Eger)]
MSC 2000:
*11B39 Special numbers, etc.
20D60 Arithmetic and combinatorial problems on finite groups
20D15 Nilpotent finite groups

Keywords: recurrences; periodic sequence; finite \$p\$-group; 3-step Fibonacci sequence; fundamental periods

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