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Nilpotent groups are not dualizable

Part of: Algebraic structures Representation theory of groups

Published online by Cambridge University Press:  09 April 2009

R. Quackenbush  and
C. S. Szabó
R. Quackenbush
Affiliation:
Department of Mathematics, University of Manitoba, Winnipeg, manitoba R3T 2N2, Canada e-mail: qubush@ccu.umanitoba.ca
C. S. Szabó
Affiliation:
Department of Algebra and Number Theory, ELTE, Budapest, Hungary e-mail: csaba@cs.elte.hu
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Abstract

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It is shown that no finite group containing a non-abelian nilpotent subgroup is dualizable. This is in contrast to the known result that every finite abelian group is dualizable (as part of the Pontryagin duality for all abelian groups) and to the result of the authors in a companion article that every finite group with cyclic Sylow subgroups is dualizable.

MSC classification

Secondary: 20D15: Nilpotent groups, $p$-groups 08A05: Structure theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 72 , Issue 2 , April 2002 , pp. 173 - 180
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Clark, D. M. and Davey, B. A., Natural dualities for the working algebraist (CUP, Cambridge, 1998).Google Scholar
[2]Davey, B. A., ‘Duality theory on ten dollars a day,’ in Algebras and orders (Monteral, PQ, 1991), NATo Adv. Sci. Inst. Ser. C Math. Phys. Sci. 389 (Kluwer Acad. Publ, Dorecht, 1993)pp. 71–111.CrossRefGoogle Scholar
[3]Davey, B. A. and Quackenbush, R. W., ‘Natural dualities for dihedral varieties’, J. Austral. Math. Soc. (Ser. A) 61 (1996), 216–228.CrossRefGoogle Scholar
[4]Quackenbush, R. and Szabó, Cs., ‘Strong duality for metacyclic groups,’ J. Austral. Math. Soc., to appear.Google Scholar