ALL GROUPS ARE OUTER AUTOMORPHISM GROUPS OF SIMPLE GROUPS

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.