Abstract
For a non-Abelian 2-generated finite group G=〈a,b〉, the Fibonacci length of G with respect to A={a,b}, denoted by LEN A (G), is defined to be the period of the sequence x 1=a,x 2=b,x 3=x 1 x 2,…,x n+1=x n−1 x n ,… of the elements of G. For a finite cyclic group C n =〈a〉, LEN A (C n ) is defined in a similar way where A={1,a} and it is known that LEN A (C n )=k(n), the well-known Wall number of n. Over all of the interesting numerical results on the Fibonacci length of finite groups which have been obtained by many authors since 1990, an intrinsic property has been studied in this paper. Indeed, by studying the family of minimal non-Abelian p-groups it will be shown that for every group G of this family, there exists a suitable generating set A′ for the derived subgroup G′ such that LEN A′(G′)|LEN A (G) where, A is the original generating set of G.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aydin, H., Smith, G.C.: Finite p-quotients of some cyclically presented groups. J. Lond. Math. Soc. 49, 83–92 (1994)
Campbell, C.M., Campbell, P.P.: The Fibonacci length of certain centro-polyhedral groups. J. Appl. Math. Comput. 19, 231–240 (2005)
Campbell, C.M., Doostie, H., Robertson, E.F.: Fibonacci length of generating pairs in groups. In: Bergum, G.A., et al. (eds.) Applications of Fibonacci Numbers, vol. 5, pp. 27–35 (1990)
Campbell, C.M., Campbell, P.P., Doostie, H., Robertson, E.F.: On the Fibonacci length of powers of dihedral groups. In: Howard, F.T. (ed.) Applications of Fibonacci Numbers, vol. 9, pp. 69–85 (2004)
Campbell, C.M., Campbell, P.P., Doostie, H., Robertson, E.F.: Fibonacci length for certain metacyclic groups. Algebra Colloq. 11, 215–222 (2004)
Campbell, P.P.: Fibonacci length and efficiency in group presentations. PhD thesis, University of St. Andrews, Scotland (2003)
Doostie, H., Campbell, C.M.: Fibonacci length of automorphism groups involving Tribonacci numbers. Vietnam J. Math. 28, 57–65 (2000)
Doostie, H., Gholamie, R.: Computing on the Fibonacci length of finite groups. Int. J. Appl. Math. 4, 149–156 (2000)
Doostie, H., Adnani, A.T.: Fibonacci length of certain nilpotent 2-groups. ACTA Math. Sin. 23(5), 879–884 (2007)
Doostie, H., Maghasedi, M.: Fibonacci length of direct products of groups. Vietnam J. Math. 33(2), 189–197 (2005)
Doostie, H., Hashemi, M.: Fibonacci lengths involving the Wall number k(n). J. Appl. Math. Comput. 20(1–2), 171–180 (2006)
GAP, GAP-Groups, Algorithms and Programming, Aachen, St. Andrews (2002)
Redlei, L.: Das schiefe Product in der Gruppentheorie Comment. Math. Helv. 20, 225–267 (1947)
Sims, C.C.: Computation with Finitely Presented Groups. Encyclopedia of Mathematics and its Applications, vol. 48. Cambridge University Press, Cambridge (1994)
Wall, D.D.: Fibonacci series modulo m. Am. Math. Mon. 67, 525–532 (1960)
Wilcox, H.J.: Fibonacci sequences of period n in groups. Fibonacci Q. 24, 356–361 (1986)
Zhang, Q., An, L., Xu, M.: Finite p-groups all of whose non-Abelian proper subgroups are metacyclic. Arch. Math. 87, 1–5 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Abdolzadeh, H., Azadi, M. & Doostie, H. A subgroup involvement of the Fibonacci length. J. Appl. Math. Comput. 32, 383–392 (2010). https://doi.org/10.1007/s12190-009-0257-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-009-0257-2