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Semidirect product groups with Abelian automorphism groups

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

M. J. Curran
M. J. Curran
Affiliation:
Department of Mathematics, University of Otago, Dunedin, New Zealand
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Abstract

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Miller's group of order 64 is a smallest example of a nonabelian group with an abelian automorphism group, and is the first in an infinite family of such groups formed by taking the semidirect product of a cyclic group of order 2m (m ≥ 3) with a dihedral group of order 8. This paper gives a method for constructing further examples of non abelian 2-groups which have abelian automorphism groups. Such a 2-group is the semidirect product of a cyclic group and a special 2-group (satisfying certain conditions). The automorphism group of this semidirect product is shown to be isomorphic to the central automorphism group of the corresponding direct product. The conditions satisfied by the special 2-group are determined by establishing when this direct product has an abelian central automorphism group.

MSC classification

Secondary: 20F28: Automorphism groups of groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 42 , Issue 1 , February 1987 , pp. 84 - 91
Copyright
Copyright © Australian Mathematical Society 1987

References

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