Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-01-13T18:09:39.298Z Has data issue: false hasContentIssue false
  • English
  • Français

The orders of the Fibonacci groups

Published online by Cambridge University Press:  14 November 2011

D. J. Seal
D. J. Seal
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge
Get access Rights & Permissions [Opens in a new window]

Synopsis

The Fibonacci groups F(r, n) have been studied by various authors, chiefly in order to determine which ones are finite. This article contains a summary of the known results about this problem, followed by some further results obtained by the author. In particular, the orders of the groups F(r, 3) for r ≡ 2 (mod 3) and F(r, 4) for r ≡ 2 (mod 4) are determined, and various other Fibonacci groups are proved infinite by methods similar to those of Chalk and Johnson.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 92 , Issue 3-4 , 1982 , pp. 181 - 192
Copyright
Copyright © Royal Society of Edinburgh 1982

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Brunner, A. M.. The determination of Fibonacci groups. Bull. Austral. Math. Soc. 11 (1974), 11–14.CrossRefGoogle Scholar
2Brunner, A. M.. On groups of Fibonacci type. Proc. Edinburgh Math. Soc. 20 (1976-1977), 211–213.CrossRefGoogle Scholar
3Campbell, C. M. and Robertson, E. F.. The orders of certain metacyclic groups. Bull. London Math. Soc. 6 (1974), 312–314.CrossRefGoogle Scholar
4Campbell, C. M. and Robertson, E. F.. Applications of the Todd-Coxeter algorithm to generalised Fibonacci groups. Proc. Roy. Soc. Edinburgh Sect. A 73 (1974), 163–166.CrossRefGoogle Scholar
5Campbell, C. M. and Robertson, E. F.. On a class of finitely presented groups of Fibonacci type. J. London Math. Soc. 11 (1975), 249–255.CrossRefGoogle Scholar
6Campbell, C. M. and Robertson, E. F.. On metacyclic Fibonacci groups. Proc. Edinburgh Math. Soc. 19 (1975), 253–256.CrossRefGoogle Scholar
7Chalk, C. P. and Johnson, D. L.. The Fibonacci groups. II. Proc. Roy. Soc. Edinburgh Sect. A 77 (1977), 79–86.CrossRefGoogle Scholar
8Conway, J. H.. Advanced problem 5327, Amer. Math. Monthly 72 (1965), 915, and solution to same, ibid. 74 (1967), 91–93.Google Scholar
9Havas, G.. Computer-aided determination of a Fibonacci group. Bull. Austral. Math. Soc. 15 (1976), 297–305.CrossRefGoogle Scholar
10Havas, G., Richardson, J. S. and Sterling, L. S.. The last of the Fibonacci groups. Proc. Roy. Soc. Edinburgh Sect. A 83 (1979), 199–203.CrossRefGoogle Scholar
11Johnson, D. L.. A note on the Fibonacci groups. Israel J. Math. 17 (1974), 277–282.CrossRefGoogle Scholar
12Johnson, D. L.. Extensions of Fibonacci groups. Bull. London Math. Soc. 7 (1974), 101–104.CrossRefGoogle Scholar
13Johnson, D. L.. Some infinite Fibonacci groups. Proc. Edinburgh Math. Soc. 19 (1975), 311–314.CrossRefGoogle Scholar
14Johnson, D. L., Wamsley, J. W. and Wright, D.. The Fibonacci groups. Proc. London Math. Soc. 29 (1974), 577–592.CrossRefGoogle Scholar
15Lyndon, R. C., unpublished proof that F(2, n) is infinite for n≧ll.Google Scholar
16Lyndon, R. C. and Schupp, P. E.. Combinatorial Group Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete 89 (Berlin: Springer-Verlag, 1977).Google Scholar