On Mennicke groups of deficiency zero. II. (English) Zbl 0806.20026
Summary: [For part I, cf. Int. J. Math. Math. Sci. 8, 821-824 (1985; Zbl 0588.20024).]
Let \(M\) be the group defined by the presentation \(\langle x,y,z \mid x^ y = x^ m,\;y^ z = y^ n,\;z^ x = z^ r\rangle\), \(m,n,r \in Z\). \(M\) is one of the few 3-generator finite groups of deficiency zero. These groups have been considered by Mennicke, Macdonald, Wamsley, Johnson and Robertson 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. We also describe the structure of some cases of \(M\).
Let \(M\) be the group defined by the presentation \(\langle x,y,z \mid x^ y = x^ m,\;y^ z = y^ n,\;z^ x = z^ r\rangle\), \(m,n,r \in Z\). \(M\) is one of the few 3-generator finite groups of deficiency zero. These groups have been considered by Mennicke, Macdonald, Wamsley, Johnson and Robertson 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. We also describe the structure of some cases of \(M\).
MSC:
| 20F05 | Generators, relations, and presentations of groups |
