Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-11T06:28:53.179Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[1]Curran, M. J., ‘A non-abelian automorphism group with all automorphisms central’, Bull. Austral. Math. Soc. 26 (1982), 393–397.CrossRefGoogle Scholar
[2]Dixon, J. D., Problems in group theory (Dover Publications, New York, 1973).Google Scholar
[3]Earnley, B. E., On finite groups whose group of automorphisms is abelian (Ph. D. thesis, Wayne State University, 1975).Google Scholar
[4]Hall, M. Jr, and Senior, J. K., The groups of order 2n, (n ≤ 6), (New York, 1964).Google Scholar
[5]Jonah, D. and Konvisser, M., ‘Some non-abelian p–group with abelian automorphism groups’, Arch. Math. 26 (1975), 131–133.CrossRefGoogle Scholar
[6]Miller, G. A., ‘A non-abelian group whose group of isomorphisms is abelian’, Messenger of Math. 43 (1913), 124–125.Google Scholar
[7]Sanders, P. R., ‘The central automorphisms of a finite group’, J. London Math. Soc. 44 (1969), 225–228.CrossRefGoogle Scholar
[8]Struik, R. R., ‘Some non-abelian 2-groups with abelian automorphism groups’, Arch. Math. 39, (1982), 299–302.CrossRefGoogle Scholar