Journal of the London Mathematical Society



ALL GROUPS ARE OUTER AUTOMORPHISM GROUPS OF SIMPLE GROUPS


MANFRED DROSTE a1 1 , MICHÈLE GIRAUDET a2 1 and RÜDIGER GÖBEL a3 1
a1 Institut für Algebra, Technische Universität Dresden, 01062 Dresden, Germany; droste@math.tu-dresden.de
a2 Département de Mathématiques, Universitè du Maine, 72085 Le Mans Cedex 09, France; giraudet@logique.jussieu.fr
a3 Fachbereich 6, Mathematik und Informatik, Universität Essen, 45117 Essen, Germany; r.goebel@uni-essen.de

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, [less-than-or-eq, slant]). However, Aut T is never simple. Following recent investigations on automorphism groups of circles, it is possible to turn (T, [less-than-or-eq, slant]) into a circle C such that Out (Aut T) [reverse congruent] Out (Aut C). The unavoidable normal subgroups in Aut T evaporate in Aut C, which is now simple, and the result follows.

(Received November 17 2000)
(Revised March 30 2001)



Footnotes

1 This work is supported by project G-0545-173.06/97 of the German-Israeli Foundation for Scientific Research & Development.