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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
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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

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