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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.
    JAIN, VIVEK K. RAI, PRADEEP K. and YADAV, MANOJ K. 2013. ON FINITE p-GROUPS WITH ABELIAN AUTOMORPHISM GROUP. International Journal of Algebra and Computation, Vol. 23, Issue. 05, p. 1063.
    Jain, Vivek K. and Yadav, Manoj K. 2012. On finite p-groups whose automorphisms are all central. Israel Journal of Mathematics, Vol. 189, Issue. 1, p. 225.
    Curran, M. J. 2006. Direct Products with Abelian Automorphism Groups. Communications in Algebra, Vol. 35, Issue. 1, p. 389.
    Szechtman, Fernando 2004. The Group of Outer Automorphisms of the Semidirect Product of the Additive Group of a Ring by a Group of Units. Communications in Algebra, Vol. 32, Issue. 1, p. 19.
    Curran, M. J. and McCaughan, D. J. 2001. CENTRAL AUTOMORPHISMS THAT ARE ALMOST INNER. Communications in Algebra, Vol. 29, Issue. 5, p. 2081.
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Semidirect product groups with Abelian automorphism groups

  • M. J. Curran (a1)
Abstract

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.

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Copyright
COPYRIGHT: © Australian Mathematical Society 1987
References
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[1]Curran, M. J., ‘A non-abelian automorphism group with all automorphisms central’, Bull. Austral. Math. Soc. 26 (1982), 393–397.
[2]Dixon, J. D., Problems in group theory (Dover Publications, New York, 1973).
[3]Earnley, B. E., On finite groups whose group of automorphisms is abelian (Ph. D. thesis, Wayne State University, 1975).
[4]Hall, M. Jr, and Senior, J. K., The groups of order 2n, (n ≤ 6), (New York, 1964).
[5]Jonah, D. and Konvisser, M., ‘Some non-abelian p–group with abelian automorphism groups’, Arch. Math. 26 (1975), 131–133.
[6]Miller, G. A., ‘A non-abelian group whose group of isomorphisms is abelian’, Messenger of Math. 43 (1913), 124–125.
[7]Sanders, P. R., ‘The central automorphisms of a finite group’, J. London Math. Soc. 44 (1969), 225–228.
[8]Struik, R. R., ‘Some non-abelian 2-groups with abelian automorphism groups’, Arch. Math. 39, (1982), 299–302.
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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