Abstract
Let G be a group and let x, y ∈ G. If every element of G can be written as a word
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.
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© 1990 Kluwer Academic Publishers
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Campbell, C.M., Doostie, H., Robertson, E.F. (1990). Fibonacci Length of Generating Pairs in Groups. In: Bergum, G.E., Philippou, A.N., Horadam, A.F. (eds) Applications of Fibonacci Numbers. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-1910-5_4
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DOI: https://doi.org/10.1007/978-94-009-1910-5_4
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