A problem about normal subgroups. (English) Zbl 0797.20027
The authors prove the following theorem. Suppose \(G\) is a finitely generated group with a proper normal subgroup of finite index. If \(G\) is isomorphic to all of its non-trivial normal subgroups then \(G\) is infinite cyclic. In a particular case this answers a question raised by P. Hall.
Reviewer: Yu.A.Bakhturin (Moskva)
MSC:
| 20E34 | General structure theorems for groups |
| 20E07 | Subgroup theorems; subgroup growth |
| 20E36 | Automorphisms of infinite groups |
Keywords:
infinite groups; finitely generated group; normal subgroup of finite index; infinite cyclicReferences:
| [1] | Hall, P., The Eulerian functions of a group, Quart. J. Math. Oxford, 7, 134-151 (1936) · JFM 62.0082.02 |
| [2] | Wiegold, J.; Wilson, J. S., Growth sequences of finitely generated groups, Arch. Math., 30, 337-343 (1978) · Zbl 0405.20042 |
| [3] | Zassenhaus, H., Beweis eines Satzes über diskrete Gruppen, Abh. Math. Sem. Univ. Hamburg, 12, 289-312 (1938) · JFM 64.0961.06 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
