Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-17T10:37:44.288Z Has data issue: false hasContentIssue false
  • English
  • Français

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; droste@math.tu-dresden.de
MICHÈLE GIRAUDET
Affiliation:
Département de Mathématiques, Universitè du Maine, 72085 Le Mans Cedex 09, France; giraudet@logique.jussieu.fr
RÜDIGER GÖBEL
Affiliation:
Fachbereich 6, Mathematik und Informatik, Universität Essen, 45117 Essen, Germany; r.goebel@uni-essen.de
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)