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On a Group Presentation Due to Fox

Published online by Cambridge University Press:  20 November 2018

C. M. Campbell  and
E. F. Robertson
C. M. Campbell
Affiliation:
Mathematical Institute, University of St. Andrews, St. Andrews, KY16 9SS, Scotland
E. F. Robertson
Affiliation:
Mathematical Institute, University of St. Andrews, St. Andrews, KY16 9SS, Scotland
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In 1956 R. H. Fox had occasion, while investigating fundamental groups of topological surfaces, to believe that the group <a, b | ab 2=b 3 a, ba 2=a 2 b> was trivial. Using the Todd-Coxeter coset enumeration algorithm a proof was obtained, see [3], and this algorithmic proof was used to produce an algebraic proof, see [2]. In [1] Benson and Mendelsohn, using a similar method to that of [2] showed that <a, b | ab n =b n+1 a, ba n =a n+1 b> is trivial. In this note we give a direct proof for the more general problem of describing the structure of the group <a, b | ab n =b ℓ a, ba n =a ℓ b>.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 19 , Issue 2 , 01 June 1976 , pp. 247 - 248
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Benson, C. T. and Mendelsohn, N. S., A calculus for a certain class of word problems in groups, J. Combinatorial Theory 1 (1966), 202–208.CrossRefGoogle Scholar
2. Campbell, C. M., Enumeration of cosets and solutions of some word problems in groups, Dissertation, McGill University, 1965.Google Scholar
3. Coxeter, H. S. M. and Moser, W. O. J., Generators and Relations for Discrete groups (Springer, Berlin, 2nd. edition 1965).Google Scholar