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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

REFERENCES

1.Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras (Interscience Publishers, 1962).Google Scholar
2.Huppert, B., Endliche Gruppen I (Springer-Verlag, 1967).CrossRefGoogle Scholar
3.Lennox, J. C. and Stonehewer, S. E., Subnormal subgroups of groups, Oxford Mathematical Monographs (Oxford, 1987).Google Scholar
4.McCaughan, D. J., Subnormality in soluble minimax groups, J. Austral. Math. Soc. 27 (1974), 113–128.CrossRefGoogle Scholar
5.Robinson, D. J. S., Finiteness conditions and generalized soluble groups, Part 1 (Springer-Verlag, 1972).CrossRefGoogle Scholar
6.Robinson, D. J. S., A course in the theory of groups (Springer-Verlag, 1982).CrossRefGoogle Scholar
7.Schenkman, E., On the norm of a group, Illinois J. Math. 4 (1960), 150–152.CrossRefGoogle Scholar
8.Segal, D., Polycyclic groups (Cambridge University Press, 1983).CrossRefGoogle Scholar
9.Wielandt, H., Über den Normalisator der Subnormalen Untergruppen, Math. Z. 69 (1958), 463–465.CrossRefGoogle Scholar