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ON THE CENTRE OF THE AUTOMORPHISM GROUP OF A GROUP

Part of: Elementary number theory Special aspects of infinite or finite groups

Published online by Cambridge University Press:  16 June 2015

M. FARROKHI D. G.  and
MOHAMMAD REZA R. MOGHADDAM
M. FARROKHI D. G.*
Affiliation:
Mathematical Science Research Unit, College of Liberal Arts, Muroran Institute of Technology, 27-1, Mizumoto, Muroran 050-8585, Hokkaido, Japan email m.farrokhi.d.g@gmail.com
MOHAMMAD REZA R. MOGHADDAM
Affiliation:
Department of Pure Mathematics and the Centre of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, Mashhad, Iran Department of Mathematics, Khayyam University, Mashhad, Iran email rezam@ferdowsi.um.ac.ir
*
✉m.farrokhi.d.g@gmail.com
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Abstract

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If the centre of a group $G$ is trivial, then so is the centre of its automorphism group. We study the structure of the centre of the automorphism group of a group $G$ when the centre of $G$ is a cyclic group. In particular, it is shown that the exponent of $Z(\text{Aut}(G))$ is less than or equal to the exponent of $Z(G)$ in this case.

Keywords

automorphism groupcentrep-groupprimitive root

MSC classification

Primary: 20F28: Automorphism groups of groups
Secondary: 11A07: Congruences; primitive roots; residue systems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 3 , December 2015 , pp. 390 - 396
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

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Deaconescu, M. and Walls, G. L., ‘On the group of automorphisms of a group’, Amer. Math. Monthly 118(5) (2011), 452–455.CrossRefGoogle Scholar
Formanek, E., ‘Fixed points and centers of automorphism groups of free nilpotent groups’, Comm. Algebra 30(2) (2002), 1033–1038.CrossRefGoogle Scholar