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Central automorphisms of finite groups

Published online by Cambridge University Press:  17 April 2009

M. J. Curran  and
D. J. McCaughan
M. J. Curran
Affiliation:
Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin, New Zealand
D. J. McCaughan
Affiliation:
Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin, New Zealand
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Abstract

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This paper considers an aspect of the general problem of how the structure of a group influences the structure of its automorphisms group. A recent result of Beisiegel shows that if P is a p-group then the central automorphisms group of P has no normal subgroups of order prime to p. So, roughly speaking, most of the central automorphisms are of p-power order. This generalizes an old result of Hopkins that if Aut P is abelian (so every automorphisms is central), then Aut P is a p-group.

This paper uses a different approach to consider the case when P is a π-group. It is shown that the central automorphism group of P has a normal. π′-subgroup only if P has an abelian direct factor whose automorphism group has such a subgroup.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 34 , Issue 2 , October 1986 , pp. 191 - 198
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Adney, J.E. and Yen, T., “Automorphisms of a p-group”. Illinois J. Math. 9 (1965), 137–143.CrossRefGoogle Scholar
[2]Beisiegel, B., “Finite p-groups with non-trivial p'-automorphisms”. Arch. Math.(Basel), 31 (1978), 209–216.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]Gorenstein, D., Finite Groups. (New York, 1968.)Google Scholar
[5]Hopkins, C., “Non-abelian groups whose groups of isomorphisms are abelian”. Ann. of Math. 29 (1927/1928), 508–520.Google Scholar
[6]Jonah, D.W. and Konvisser, M. W., “Some non-abelian p-groups with abelian automorphisms groups”. Arch. Math. (Basel), 26 (1975), 131–133.CrossRefGoogle Scholar
[7]Miller, G.A., “A non-abelian group whose group of isomorphisms is abelian”. Messenger of Math. 43 (1913), 124–125.Google Scholar
[8]Sanders, P.R., “The central automorphisms of a finite group”. J. London Math. Soc. 44 (1969), 225–228.Google Scholar
[9]Struik, R.R., “Some non-abelian 2-groups with abelian automorphism groups”. Arch. Math. (Basel) 39 (1982), 299–302.CrossRefGoogle Scholar
[10]Ying, J.H., “On finite groups whose automorphisms groups are nilpotent”. Arch. Math (Basel) 29 (1977), 41–44.CrossRefGoogle Scholar