Question

(5) Fibonacci sequences in groups. The Fibonacci numbers F, are defined recursively by Fo = 0, Fi-1, and Fn Fn-1 + Fn-2 for n > 2. The definition of this sequence only depends on a binary operation. Since every group comes with a binary operation, we can define Fibonacc type sequences in any group. Let G be a group, and define the sequence (n in G as follows: Let ao, ai be elements of G, and define fo-ao fa and fnan-1an-2 forn 2 In his paper "Fibonacci Series Modulo m" (American Mathematical Monthly, Vol. 67, 1960), D.D. Wall writes the following about his introduction to these sequences: The problem arose in connection with a method for generating random numbers, but it turned out to be unexpectedly intricate, and so quickly became of interest in its own right." In an interesting application of Fibonacci sequences in groups, lannis Xenakis in Formal ized Music, Indiana University Press, 1971) uses Fibonacci sequences in groups to create "Fibonacci motions," which are sequences of musical properties such as pitch, volume, and timbre that give the composition its framework. We will see in this problem that these Fi bonacci sequences become intricate quite quickly (a) Explain why fn-Fn if G Z, ao 0, and a1 (b) Find the elements in Un } if G-Z3, ao = [1] and a 1-[2] (c) Some of the sequences are periodic-that is, the same list of elements repeats in the same order. The number of elements in one cycle is called the period of the sequence (G) If a is a non-identity element of a group with identity e and ae, what is the (ii) Ifa is a non-identity element of a group with identity e and a e, what is the (ii) If a is a non-identity element of a group with identity e and ae but a2 e, (iv) If a is a non-identity element of a group with identity e ae, what is the period of the sequence that begins aa...? period of the sequence that begins a a ...? what is the period of the sequence that begins ca . . .? period of the sequence that begins a . . .?

Answer 1

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