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The Structure of the (Outer) Automorphism Group of a Bieberbach Group

Published online by Cambridge University Press:  04 December 2007

Wim Malfait  and
Andrzej Szczepański
Wim Malfait
Affiliation:
Department of Mathematics, Katholieke Universiteit Leuven, Campus Kortrijk, Universitaire Campus, Etienne Sabbelaan 53, B-8500 Kortrijk, Belgium. e-mail: wim.malfait@kulak.ac.be
Andrzej Szczepański
Affiliation:
Department of Mathematics, Katholieke Universiteit Leuven, Campus Kortrijk, Universitaire Campus, Etienne Sabbelaan 53, B-8500 Kortrijk, Belgium. e-mail: wim.malfait@kulak.ac.be
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Abstract

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Referring to Tits′ alternative, we develop a necessary and sufficient condition to decide whether the normalizer of a finite group of integral matrices is polycyclic-by-finite or is containing a non-Abelian free group. This result is of fundamental importance to conclude whether the (outer) automorphism group of a Bieberbach group is polycyclic-by-finite or has a non-cyclic free subgroup.

Keywords

(outer) automorphism groupBieberbach groupdicyclic groupfinite group of integral matricespolycyclic-by-finite group(ir)reducible representationSchur indexunit group of an order.
Type
Research Article
Information
Compositio Mathematica , Volume 136 , Issue 1 , March 2003 , pp. 89 - 101
Copyright
© 2003 Kluwer Academic Publishers