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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

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