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On Mennicke Groups of Deficiency Zero II

Published online by Cambridge University Press:  20 November 2018

Muhammad A. Albar  and
Abdul-Aziz A. Al-Shuaibi
Muhammad A. Albar
Affiliation:
Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia
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Abstract

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Let M be the group defined by the presentation 〈 x, y, z | xy = xm, yz = yn, zx = zr〉,m,n,r ∊ Z. M is one of the few 3-generator finite groups of deficiency zero. These groups have been considered by Mennicke [3], Macdonald, Wamsley [10], Johnson and Robertson [7], and Albar. Properties like the order of M, the nilpotency and solvability were studied. In this paper we give a better upper bound for M than the one given by Johnson and Robertson [7]. We also describe the structure of some cases of M.

Keywords

20F05Presentation of a groupsplit extensionTietze transformationReidemeister- Schreier process
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 34 , Issue 3 , 01 September 1991 , pp. 289 - 293
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Al-Shuaibi, A. A., On Mennicke groups of deficiency zero. M.Sc. Thesis, KFUPM, 1987.Google Scholar
2. Albar, M. A., On presentations of group extensions, Comm. Algebra 12 (1984), 2967–2975.Google Scholar
3. Albar, M. A., On Mennicke groups of deficiency zero I, Internat. J. Math. Math. Sci. (4)8 (1985), 821–824.Google Scholar
4. Fluch, W., A generalizedHigman group, Nederl. Akad. Wetensch. Indag. Math. 44 (1982), 153–166.Google Scholar
5. Higman, G., A finitely generated infinite simple group, J. London Math. Soc. 26 (1951), 61–64.Google Scholar
6. Johnson, D. L., Topics in the Theory of Group Presentation. London Math. Soc. Lecture Notes Series 42, Cambridge University Press, 1980.Google Scholar
7. Johnson, D. L. and Robertson, E. F., Finite Groups of Deficiency Zero. In Homological Group Theory (ed. C.T.C. Wall), Cambridge University Press, 1979.Google Scholar
8. Mennicke, J., Einige endliche gruppen mit drei erzeugenden und drei relationen, Arch. Math. 10 (1959), 409–18.Google Scholar
9. Post, M. J., Finite three generator groups with zero deficiency, Comm. Algebra 6 (1978), 1289–1296.Google Scholar
10. Wamsley, J. W., The deficiency of finite groups. Ph.D. Thesis, University of Queensland, 1968.Google Scholar