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The Wielandt subgroup of metacyclic p-groups

  • Elizabeth A. Ormerod (a1)

Abstract

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.

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[1]Camina, A.R., ‘The Wielandt length of finite groups’, J. Algebra 15 (1970), 142148.
[2]King, Bruce W., ‘Presentations of metacyclic groups’, Bull. Austral. Math. Soc. 8 (1973), 103131.
[3]Schenkman, E., ‘On the norm of a group’, Illinois J. Math. 4 (1960), 150152.
[4]Newman, M.F. and Mingyao, Xu, ‘Metacyclic groups of prime-power order’, Adv. in Math. 17 (1988), 106170.
[5]Wielandt, Helmut, ‘Über den Normalisator der subnormalen Untergruppen’, Math. Z. 69 (1958), 463465.
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