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The wielandt series of metabelian groups

Published online by Cambridge University Press:  17 April 2009

C.J.T. Wetherell
C.J.T. Wetherell
Affiliation:
Mathematical Sciences Institute, ANU Canberra, ACT 0200, Australia
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Abstract

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The Wielandt subgroup of a group is the intersection of the normalisers of its subnormal subgroups. It is non-trivial in any finite group and thus gives rise to a series whose length provides a measure of the complexity of the group's subnormal structure. In this paper results of Ormerod concerning the interplay between the Wielandt series and upper central series of metabelian p-groups, p odd, are extended to the class of all odd order metabelian groups. These extensions are formulated in terms of a natural generalization of the upper central series which arises from Casolo's strong Wielandt subgroup, the intersection of the centralisers of a group's nilpotent subnormal sections.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 67 , Issue 2 , April 2003 , pp. 267 - 276
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Ali, A., ‘On the Wielandt length of a finite supersoluble group’, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), 1217–1226.CrossRefGoogle Scholar
[2]Bryce, R.A. and Cossey, J., ‘The Wielandt subgroup of a finite soluble group’, J. London Math. Soc. (2) 40 (1989), 244–256.CrossRefGoogle Scholar
[3]Camina, A.R., ‘The Wielandt length of finite groups’, J. Algebra 15 (1970), 142–148.CrossRefGoogle Scholar
[4]Casolo, C., ‘Wielandt series and defects of subnormal subgroups in finite soluble groups’, Rend. Sem. Mat. Univ. Padova 87 (1992), 93–104.Google Scholar
[5]Doerk, K. and Hawkes, T.O., Finite soluble groups (Walter de Gruyter, Berlin, 1992).CrossRefGoogle Scholar
[6]Ormerod, E.A., ‘The Wielandt subgroup of metacyclic p-groups’, Bull. Austral. Math. Soc. 42 (1990), 499–510.CrossRefGoogle Scholar
[7]Ormerod, E.A., ‘Groups of Wielandt length two’, Math. Proc. Cambridge Philos. Soc. 110 (1991), 229–244.CrossRefGoogle Scholar
[8]Ormerod, E.A., ‘On the Wielandt length of metabelian p-groups’, Arch. Math. (Basel) 57 (1991), 212–215.CrossRefGoogle Scholar
[9]Ormerod, E.A., ‘Some p-groups of Wielandt length three’, Bull. Austral. Math. Soc. 58 (1998), 121–136.CrossRefGoogle Scholar
[10]Ormerod, E.A., ‘A note on the Wielandt subgroup of a metabelian p-group’, Comm. Algebra 27 (1999), 621–627.CrossRefGoogle Scholar
[11]Schenkman, E., ‘On the norm of a group’, Illinois J. Math. 4 (1960), 150–152.CrossRefGoogle Scholar
[12]Wetherell, C.J.T., Subnormal structure of finite soluble groups, (Ph.D. thesis) (Australian National University, ACT, Australia, 2001). http://thesis.anu.edu.au.Google Scholar
[13]Wielandt, H., ‘Über den Normalisator der subnormalen Untergruppen’, Math. Z. 69 (1958), 463–465.CrossRefGoogle Scholar