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Finite groups of deficiency zero involving the Lucas numbers

Published online by Cambridge University Press:  20 January 2009

C. M. Campbell ,
E. F. Robertson  and
R. M. Thomas
C. M. Campbell
Affiliation:
Mathematical Institute, University of St Andrews, St Andrews KY16 9SS
E. F. Robertson
Affiliation:
Mathematical Institute, University of St Andrews, St Andrews KY16 9SS
R. M. Thomas
Affiliation:
Department of Computing Studies, University of Leicester, Leicester LE1 7RH
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Abstract

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In this paper, we investigate a class of 2-generator 2-relator groups G(n) related to the Fibonacci groups F(2,n), each of the groups in this new class also being defined by a single parameter n, though here n can take negative, as well as positive, values. If n is odd, we show that G(n) is a finite soluble group of derived length 2 (if n is coprime to 3) or 3 (otherwise), and order |2n(n + 2)gnf(n, 3)|, where fn is the Fibonacci number defined by f0=0,f1=1,fn+2=fn+fn+1 and gn is the Lucas number defined by g0 = 2, g1 = 1, gn+2 = gn + gn+1 for n≧0. On the other hand, if n is even then, with three exceptions, namely the cases n = 2,4 or –4, G(n) is infinite; the groups G(2), G(4) and G(–4) have orders 16, 240 and 80 respectively.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 33 , Issue 1 , February 1990 , pp. 1 - 10
Copyright
Copyright © Edinburgh Mathematical Society 1990

References

REFERENCES

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