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The Wielandt subgroup of a polycyclic group

Published online by Cambridge University Press:  18 May 2009

John Cossey
John Cossey
Affiliation:
Department of Mathematics, Australian National University, GPO Box 4, Canberra, ACT 2601, Australia
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The purpose of this paper is to establish some basic properties of the Wielandt subgroup of a polycyclic group. The Wielandt subgroup of a group G is defined to be the intersection of the normalisers of all the subnormal subgroups of G and is denoted by ω(G). In 1958 Wielandt [9] showed that any minimal normal subgroup with the minimum condition on subnormal subgroups is contained in the Wielandt subgroup: it follows that the Wielandt subgroup of an artinian group is nontrivial. In contrast, the Wielandt subgroup of a polycyclic group can be trivial; an easy example is given by the infinite dihedral group. We will show that the Wielandt subgroup of a polycyclic group is close to being central.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 33 , Issue 2 , May 1991 , pp. 231 - 234
Copyright
Copyright © Glasgow Mathematical Journal Trust 1991

References

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