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The Wielandt subgroup of metacyclic p-groups

Published online by Cambridge University Press:  17 April 2009

Elizabeth A. Ormerod
Elizabeth A. Ormerod
Affiliation:
Department of Mathematics Faculty of Science, The Australian National University, GPO Box 4 Canberra ACT 2601, Australia
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Abstract

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The Wielandt subgroup is the intersection of the normalisers of all the subnormal subgroups of a group. For a finite group it is a non-trivial characteristic subgroup, and this makes it possible to define an ascending normal series terminating at the group. This series is called the Wielandt series and its length determines the Wielandt length of the group. In this paper the Wielandt subgroup of a metacyclic p–group is identified, and using this information it is shown that if a metacyclic p–group has Wielandt length n, its nilpotency class is n or n + 1.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 42 , Issue 3 , December 1990 , pp. 499 - 510
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Camina, A.R., ‘The Wielandt length of finite groups’, J. Algebra 15 (1970), 142–148.CrossRefGoogle Scholar
[2]King, Bruce W., ‘Presentations of metacyclic groups’, Bull. Austral. Math. Soc. 8 (1973), 103–131.CrossRefGoogle Scholar
[3]Schenkman, E., ‘On the norm of a group’, Illinois J. Math. 4 (1960), 150–152.CrossRefGoogle Scholar
[4]Newman, M.F. and Mingyao, Xu, ‘Metacyclic groups of prime-power order’, Adv. in Math. 17 (1988), 106–170.Google Scholar
[5]Wielandt, Helmut, ‘Über den Normalisator der subnormalen Untergruppen’, Math. Z. 69 (1958), 463–465.CrossRefGoogle Scholar