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A Class of Three-Generator, Three-Relation, Finite Groups

Published online by Cambridge University Press:  20 November 2018

J. W. Wamsley
J. W. Wamsley*
Affiliation:
University of Queensland, St. Lucia, Brisbane The Flinders University of South Australia, Bedford Park, South Australia
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Mennicke (2) has given a class of three-generator, three-relation finite groups. In this paper we present a further class of three-generator, threerelation groups which we show are finite.

The groups presented are defined as:

with α|γ| ≠ 1, β|γ| ≠ 1, γ ≠ 0.

We prove the following result.

THEOREM 1. Each of the groups presented is a finite soluble group.

We state the following theorem proved by Macdonald (1).

THEOREM 2. G1(α, β, 1) is a finite nilpotent group.

1. In this section we make some elementary remarks.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 1 , 01 February 1970 , pp. 36 - 40
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Macdonald, I. D., On a class of finitely presented groups, Can. J. Math. 14 (1962), 602–613.Google Scholar
2. Mennicke, J., Einige endliche Gruppen mit drei Erzeugenden und drei Relationen, Arch. Math. 10 (1959), 409–418.Google Scholar