The automorphism group of a resplendent model (Q453194)

This short paper is devoted to the proof of one theorem: If \(M\) is an infinite resplendent structure for some finite first-order language, then the theory of the automorphism group of \(M\) is undecidable. The result is closely related to the result of \textit{V. V. Bludov, M. Giraudet, A. M. W. Glass} and \textit{G. Sabbagh} [``Automorphism groups of models of first order theories'', in: Models, modules and abelian groups. In memory of A. L. S. Corner. Berlin: Walter de Gruyter. 325--328 (2008; Zbl 1188.03025)] stating that if \(T\) is an arbitrary first-order theory having an infinite model, then it has a model whose automorphism group is undecidable. In this result the language is not assumed to be finite. Schmerl's proof relies on the proof of Bludov et~al.~and on a general result of Schmerl on automorphism groups of countable recursively saturated structures.