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ALL GROUPS ARE OUTER AUTOMORPHISM GROUPS OF SIMPLE GROUPS

Published online by Cambridge University Press:  30 January 2002

MANFRED DROSTE ,
MICHÈLE GIRAUDET  and
RÜDIGER GÖBEL
MANFRED DROSTE
Affiliation:
Institut für Algebra, Technische Universität Dresden, 01062 Dresden, Germany; [email protected]
MICHÈLE GIRAUDET
Affiliation:
Département de Mathématiques, Universitè du Maine, 72085 Le Mans Cedex 09, France; [email protected]
RÜDIGER GÖBEL
Affiliation:
Fachbereich 6, Mathematik und Informatik, Universität Essen, 45117 Essen, Germany; [email protected]
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Abstract

It is shown that each group is the outer automorphism group of a simple group. Surprisingly, the proof is mainly based on the theory of ordered or relational structures and their symmetry groups. By a recent result of Droste and Shelah, any group is the outer automorphism group Out (Aut T) of the automorphism group Aut T of a doubly homogeneous chain (T, [les ]). However, Aut T is never simple. Following recent investigations on automorphism groups of circles, it is possible to turn (T, [les ]) into a circle C such that Out (Aut T) [bcong ] Out (Aut C). The unavoidable normal subgroups in Aut T evaporate in Aut C, which is now simple, and the result follows.

Type
Research Article
Information
Journal of the London Mathematical Society , Volume 64 , Issue 3 , December 2001 , pp. 565 - 575
Copyright
London Mathematical Society 2001

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