Abstract
The problems discussed in this paper arise from the theory of group presentations. In this section, we give a brief review of this subject, or, at least, those aspects of it which are relevant to the present paper. In the remaining sections we discuss links, occurring in our work over a number of years, between this topic and the Fibonacci and Lucas sequences of numbers (f n ) and (g n )
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Brunner, A. M. “The determination of Fibonacci groups.” Bull. Austral. Math. Soc. 11 (1974): pp 11–14
Brunner, A. M. “On groups of Fibonacci type.” Proc. Edinburgh Math. Soc 20 (1977): pp 211–213
Campbell, C. M., Coxeter, H. S. M. and Robertson, E. F. “Some families of finite groups having two generators and two relations.” Proc. Roy. Soc. London 357A (1977): pp 423–438
Campbell, C. M. and Robertson, E. F. “The orders of certain metacyclic groups.” Bull. London Math. Soc. 6 (1974): pp 312–314
Campbell, C. M. and Robertson, E. F. “Applications of the Todd-Coxeter algorithm to generalized Fibonacci groups.” Proc. Roy. Soc. Edinburgh 73A (1974/5): pp 163–166
Campbell, C. M. and Robertson, E. F. “On metacyclic Fibonacci groups.” Proc. Edinburgh Math. Soc 19 (1975): pp 253–256
Ca mpbeli, C. M. and Robertson, E. F. “A note on Fibonacci type groups.” Canad. Math. Bull 18 (1975): pp 173–175
Campbell, C. M. and Robertson, E. F. “On a class or finitely presented groups of Fibonacci type.” J. London Math. Soc 11 (1975): pp 249–255
Campbell, C. M. and Robertson, E. F. “Classes of groups related to Fa,b,c.” Proc. Roy. Soc. Edinburgh 78A (1978): pp 209–218
Campbell, C. M. and Robertson, E. F. “Deficiency zero groups involving Fibonacci and Lucas numbers.” Proc. Roy. Soc. Edinburgh 81 A(1978): pp 273–286
Campbell, C. M. and Robertson, E. F. “Groups related to Fa,b,c involving Fibonacci numbers.” in The Geometric Vein’, edited by C. Davis, B. Grünbaum and F. A. Sherk, Springer Verlag (1982): pp 569–576
Campbell, C. M. and Robertson, E. F. “Some problems in group presentations.” J. Korean Math. Soc 19 (1983): pp 123–128
Campbell, C. M. and Thomas, R. M. “On infinite groups of Fibonacci type.” Proc. Edinburgh Math. Soc 29 (1986): pp 225–232
Chalk, C. P. and Johnson, D. L. “The Fibonacci groups, II.” Proc. Roy. Soc. Edinburgh 77A (1977): pp 79–86
Conway, J. H. “Advanced problem 5327.” Amer. Math. Monthly 72 (1965): p 915
Conway, J. H. et al., “Solution to advanced problem 5327.” Amer. Math. Monthly 74 (1967): pp 91–93
Havas, G. “Computer aided determination of a Fibonacci group.” Bull. Austral. Math. Soc 15 (1976) 297–305
Havas, G., Richardson, J. S. and Sterling, L. S. “The last of the Fibonacci groups.” Proc. Roy. Soc. Edinburgh 83A (1979): pp 199–203
Johnson, D. L. “Some infinite Fibonacci groups.” Proc. Edinburgh Math. Soc 19 (1975): pp 311–314
Johnson, D. L. “Presentations of groups.” L. M. S. Lecture Notes 22, Cambridge Universtty Press (1976)
Johnson, D. L. “Topics in the theory of group presentations.” L. M. S. Lecture Notes 42, Cambrldge University Press (1980)
Johnson, D. L. and Robertson, E. F. “Finite groups of deficiency zero.” in ‘Norrological group theory’, edited by C. T C. Wall, L. M. S. Lecture Notes 36, Cambridge University Press (1979): pp 275–289
Johnson, D. L., Wamsley, J. W. and Wright, D. “The Fibonacci groups.” Proc. London Math. Soc 29 (1974): pp 577–592
Lyndon, R. C., unpublished
Lyndon, R. C. and Schupp, P. E. “Combinatorial group theory.” Ergebnisse der Mathematik und ihrer Grenzgebiete 89, Sprtnger Verlag (1977)
Magnus, W., Karrass, A. and Solitar, D. “Combinatorial group theory.” Dover (1976)
Neubuser, J. “An elementary introduction to coset table methods in computational group theory.” in ‘Groups — St. Andrews 1981’, eallted by C. M. Campbell and E. F. Robertson, L. M. S. Lecture Notes 71, Cambridge University Press (1982): pp 1–45
Rabin, M. O. “Recursive unsolvability of group theoretic problems.” Ann. of Math 67 (1958): pp 172–194
Rotman, J. J. “The theory of groups.” Allyn and Bacon (1965)
Thomas, R. M. “Some infinite Fibonacci groups.” Bull. London Math. Soc 15 (1983): pp 384–386
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1988 Springer Science+Business Media New York
About this chapter
Cite this chapter
Campbell, C.M., Robertson, E.F., Thomas, R.M. (1988). Fibonacci Numbers and Groups. In: Philippou, A.N., Horadam, A.F., Bergum, G.E. (eds) Applications of Fibonacci Numbers. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7801-1_6
Download citation
DOI: https://doi.org/10.1007/978-94-015-7801-1_6
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-8447-7
Online ISBN: 978-94-015-7801-1
eBook Packages: Springer Book Archive