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Deficiency zero groups involving Fibonacci and Lucas numbers*

Published online by Cambridge University Press:  14 November 2011

C. M. Campbell  and
E. F. Robertson
C. M. Campbell
Affiliation:
Mathematical Institute, University of St Andrews
E. F. Robertson
Affiliation:
Mathematical Institute, University of St Andrews
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Synopsis

Two closely related classes X(n) and Y(n) of two generator two relation groups are studied. The group presentations arise from an investigation of a Fibonacci type group of order 1512. The Reidemeister-Schreier algorithm is used to show that the groups X(n) are finite and not metabelian. The orders of these groups are determined and shown to be divisible by powers of Fibonacci numbers or by powers of Lucas numbers. In addition these groups add to the relatively few examples of non-metabelian two generator two relation groups whose orders are known precisely.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 81 , Issue 3-4 , 1978 , pp. 273 - 286
Copyright
Copyright © Royal Society of Edinburgh 1978

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References

1Beetham, M. J. and Campbell, C. M.A note on the Todd-Coxeter coset enumeration algorithm. Proc. Edinburgh Math. Soc. 20 (1976), 73–79.Google Scholar
2Brunner, A. M.On groups of Fibonacci type. Proc. Edinburgh Math. Soc. 20 (1977), 211–213.Google Scholar
3Campbell, C. M. and Robertson, E. F.Applications of the Todd-Coxeter algorithm to generalised Fibonacci groups. Proc. Roy. Soc. Edinburgh Sect. A. 73 (1975), 163–166.CrossRefGoogle Scholar
4Campbell, C. M. and Robertson, E. F.On a class of finitely presented groups of Fibonacci type. J. London Math. Soc. 11 (1975), 249–255.Google Scholar
5Campbell, C. M. and Robertson, E. F.A note on Fibonacci type groups. Canad. Math. Bull. 18 (1975), 173–175.CrossRefGoogle Scholar
6Campbell, 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 Ser. A 357 (1977), 423–438.Google 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.Solution to advanced problem 5327. Amer. Math. Monthly 74 (1967), 91–93.CrossRefGoogle Scholar
9Coxeter, H. S. M.Introduction to Geometry, 2nd edn (New York: Wiley, 1969).Google Scholar
10Coxeter, H. S. M. and Moser, W. O. J.Generators and Relations for Discrete Groups, 3rd edn (Berlin: Springer, 1972).Google Scholar
11Hardy, G. H. and Wright, E. M.An Introduction to the Theory of Numbers, 4th edn (Oxford Univ. Press, 1960).Google Scholar
12Johnson, D. L.A note on the Fibonacci groups. Israel J. Math. 17 (1974), 277–282.CrossRefGoogle Scholar
13Johnson, D. L.Some infinite Fibonacci groups. Proc. Edinburgh Math. Soc. 19 (1975), 311–314.Google Scholar
14Johnson, D. L. and Mawdesley, H.Some groups of Fibonacci type. J. Austral. Math. Soc. 20 (1975), 199–204.CrossRefGoogle Scholar
15Johnson, D. L., Wamsley, J. W. and Wright, D.The Fibonacci groups. Proc. London Math. Soc. 29 (1974), 577–592.Google Scholar
16Macdonald, I. D.On a class of finitely presented groups. Canad. J. Math. 14 (1962), 602–613.Google Scholar
17Wamsley, J. W.A class of two generator two relation finite groups. J. Austral. Math. Soc. 14 (1972), 38–40.Google Scholar
18Wamsley, J. W.A class of finite groups with zero deficiency. Proc. Edinburgh Math. Soc. 19 (1974), 25–29.CrossRefGoogle Scholar