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Remarks on a class of 2-generator groups of deficiency zero

Published online by Cambridge University Press:  09 April 2009

C. M. Campbell  and
E. F. Robertson
C. M. Campbell
Affiliation:
Mathematical Institute University of St. AndrewsSt. Andrews, Fife Scotland
E. F. Robertson
Affiliation:
Mathematical Institute University of St. AndrewsSt. Andrews, Fife Scotland
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Let G be a finitely presented group. A finite presentation P of G is said to have defiency m – n if it defines G with m generators and n relations. The deficiency of G is the maximum of the deficiencies of all the finite presentations P of G. If G is finite the deficiency of G is less than or equal to zero. The only finite two generator groups of deficiency zero that are known are certain metacyclic groups given by Wamsley (1970), a class of nilpotent groups given by Macdonald in (1962) and a class of groups given by Wamsley (1972).

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 19 , Issue 3 , May 1975 , pp. 297 - 305
Copyright
Copyright © Australian Mathematical Society 1975

References

Beetham, M. J. (unpublished), ‘A programme for the Todd-Coxeter coset enumeration algorithm’.Google Scholar
Beetham, M. J. and Campbell, C. M. (to appear), ‘A note on the Todd-Coxeter coset enumeration algorithm’.Google Scholar
Benson, C. T. and Mendelsohn, N. S. (1966), ‘A calculus for a certain class of word problems in groups’, J. Combinatorial Theory 1, 202–208.CrossRefGoogle Scholar
Campbell, C. M. (1969), ‘Some examples using coset enumeration’, in Computational Problems in Abstract Algebra, edited by Leech, J. (Pergamon, Oxford, 1969)Google Scholar
Coxeter, H. S. M. and Moser, W. O. J. (1972), Generators and Relations for Discrete Groups (Springer, Berlin, 3rd. ed. 1972).Google Scholar
Macdonald, I. D. (1962), ‘On a class of finitely presented groups’, Canad. J. Math. 14, 602–613.Google Scholar
Wamsley, J. W. (1970), ‘The deficiency of metacyclic groups’, Proc. Amer. Math. Soc. 24, 724–726.Google Scholar
Wamsley, J. W. (1972) ‘A class of two generator two relation finite groups’, J. Austral. Math. Soc. 14, 38–40.CrossRefGoogle Scholar
Wilde, N. W. G. (1967), ‘Benson Mendelsohn algorithm for certain word problems in groups. I.B.M. Contributed Program Library, No. 42.0.001.Google Scholar