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Some p-groups of Weilandt length three

Published online by Cambridge University Press:  17 April 2009

Elizabeth A. Ormerod
Elizabeth A. Ormerod
Affiliation:
Mathematics Department, School of Mathematical Sciences, The Australian National University, Canberra ACT 0200, Australia
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Abstract

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Finite p-groups of Wielandt length 1 are groups in which every subgroup is normal and are Dedekind groups. When the prime is odd therefore a finite p-group of Wielandt length 1 is Abelian. For an odd prime, a finite p-group of Wielandt length 2 has nilpotency class at most 3 and for such a goup to have class 3 there must be a 2-generator subgroup of this class. In this paper it is shown that for any prime p > 3 a finite p-group of Wielandt length 3 has nilpotency class at most 4, and for such a group to have class 4 there must be a 2-generator subgroup with this class. Two families of p-groups of Wielandt length 3 are described. One is a family of 3-generator groups with the property that each group modulo its Wielandt subgroup has class 2, the other is a family of 2-generator groups with the property that each group modulo its Wielandt subgroup has class 3.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 58 , Issue 1 , August 1998 , pp. 121 - 136
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Ali, A., Supersoluble groups of Wielandt length two, Ph. D. Thesis (Australian National University, Australia, 1997).Google Scholar
[2]Cooper, C., ‘Power automorphisms of a group’, Math. Z. 107 (1968), 335–356.CrossRefGoogle Scholar
[3]Hobby, C., ‘Finite groups with normal normalizer’, Canad. J. Math, 20 (1968), 1256–1260.CrossRefGoogle Scholar
[4]Mahdavi, S.K., ‘A classification of 2-generator 2-groups with many subgroups 2-subnormal’, Comm. Algebra 15 (1987), 713–750.CrossRefGoogle Scholar
[5]Ormerod, E.A., ‘On the Wielandt length of metabelian p-groups’, Arch. Math. 57 (1991), 212–215.CrossRefGoogle Scholar
[6]Ormerod, E.A., ‘Groups of Wielandt length two’, Math. Proc. Cambridge Phil. Soc. 110 (1991), 229–244.CrossRefGoogle Scholar
[7]Schenkman, E., ‘On the norm of a group’, Illinois J. Math. 4 (1960), 150–152.CrossRefGoogle Scholar
[8]Wielandt, H., ‘Über den Normalisator der subnormalen Untergruppen.’, Math. Z. 69 (1958), 463–465.CrossRefGoogle Scholar