Atlas: Any group is the outer automorphism group of a simple group by Manfred Droste

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Sixth Annual Conference in Ordered Algebraic Structures
March 5-8, 2003
Department of Mathematics, Vanderbilt University
Nashville, TN, USA

Organizers
Jorge Martinez, University of Florida and Constantine Tsinakis, Vanderbilt University

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Any group is the outer automorphism group of a simple group
by
Manfred Droste
University of Dresden, GERMANY
Coauthors: S. Shelah (resp.), M. Giraudet, R. Goebel

It is shown that each group is the outer automorphism group of a simple group. Surprisingly, our proof is mainly based on the theory of ordered structures and their lattice-ordered automorphism groups. In fact, we first show that any group G arises as the outer automorphism group of the automorphism group Aut T of a doubly homogeneous chain T. Here, previously, by a result of C. Holland, only the trivial group, Z_2, and V_4 had been realized in this way. However, this l-group, Aut T, is never simple. Therefore, we 'bend' the chain into a circle C, such that Out(Aut T) is isomorphic to Out(Aut C). The unavoidable normal subgroups in Aut T evaporate in Aut C, which is now simple, and the result follows.

Date received: December 7, 2002