Fibonacci Length of Generating Pairs in Groups

C. M. Campbell, H. Doostie and E. F. Robertson

A chapter in Applications of Fibonacci Numbers, 1990, pp 27-35 from Springer

Abstract: Abstract Let G be a group and let x, y ∈ G. If every element of G can be written as a word (1) $${x^{{\alpha _1}}}{y^{{\alpha _2}}}{x^{{\alpha _3}}} \ldots {x^{{\alpha _{n - 1}}}}{y^{{\alpha _n}}}$$ where αi ∈ ℤ, 1 ≤ i ≤ n, then we say that x and y generate G and that G is a 2-generator group. Although cyclic groups are 2-generator groups according to this definition we are only interested here in 2-generator groups which cannot be generated by a single element. Even among finite groups G many are not 2-generator groups; for example the abelian group of order 8 in which every element has order 2 cannot be 2-generated since given any pair of distinct non-trivial elements x, y there are only 4 words given by expressions of the form (1). However, many groups are 2-generated, in particular finite simple groups.

Keywords: Abelian Group; Finite Group; Generate Pair; Simple Group; Homomorphic Image (search for similar items in EconPapers)
Date: 1990
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-009-1910-5_4

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DOI: 10.1007/978-94-009-1910-5_4

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