Hostname: page-component-7bb8b95d7b-l4ctd Total loading time: 0 Render date: 2024-09-22T19:29:25.573Z Has data issue: false hasContentIssue false
  • English
  • Français

On torsion-free hypercentral groups with all subgroups subnormal

Published online by Cambridge University Press:  18 May 2009

Howard Smith
Howard Smith
Affiliation:
Department of Mathematics Bucknell, University Lewisburg, Pennsylvania 17837, USA
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

There is no example known of a non-nilpotent, torsion-free group which has all of its subgroups subnormal. It was proved in [3] that a torsion-free solvable group with all of its proper subgroups subnormal and nilpotent is itself nilpotent, but that seems to be the only published result in this area which is concerned specifically with torsion-free groups. Possibly the extra hypothesis that the group be hypercentral is sufficient to ensure nilpotency, though this is certainly not the case for groups with torsion, as was shown in [7]. The groups exhibited in that paper were seen to have hypercentral length ω + 1, and we know from [8] that further restricting the hypercentral length can lead to some positive results. Here we shall prove the following theorem.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 31 , Issue 2 , May 1989 , pp. 193 - 194
Copyright
Copyright © Glasgow Mathematical Journal Trust 1989

References

1.Brookes, C. J. B., Groups with every subgroup subnormal, Bull. London Math. Soc. 15 (1983), 235–238.CrossRefGoogle Scholar
2.Hall, P., The Edmonton notes on nilpotent groups, Q.M.C. Mathematics Notes (1979 edition).Google Scholar
3.Heineken, H. and Mohamed, I. J., A group with trivial centre satisfying the normaliser condition, J. Algebra 19 (1968), 368–376.CrossRefGoogle Scholar
4.Robinson, D. J. S., Finiteness conditions and generalised soluble groups (2 vol.), (Springer, 1972).Google Scholar
5.Robinson, D. J. S., A course in the theory of groups, Graduate Texts in Mathematics 80 (Springer, 1982).CrossRefGoogle Scholar
6.Segal, D., Poly cyclic groups, Cambridge Tracts in Mathematics 82 (Cambridge University Press, 1983).CrossRefGoogle Scholar
7.Smith, H., Hypercentral groups with all subgroups subnormal, Bull. London Math. Soc. 15 (1983), 229–234.CrossRefGoogle Scholar
8.Smith, H., Hypercentral groups with all subgroups subnormal II, Bull. London Math. Soc. 18 (1986), 343–348.CrossRefGoogle Scholar