A note on automorphism groups of countably infinite structures
Let \(\Omega\) be a countably infinite set. A relation on \(\Omega\) is just a subset of \(\Omega\) k for some integer \(k\geq 0\). The author defines a subgroup of the symmetric group Sym(\(\Omega)\) to be closed if it consists of all permutations which leave invariant the relations in some given collection of relations. Equivalently, G is closed if whenever \(x\in Sym(\Omega)\) has the property: (*) for each finite subset \(\Delta\) of \(\Omega\), there exists \(y\in G\) such that \(xy^{-1}\) fixes \(\Delta\) pointwise, then \(x\in G.\) The main result of the present paper is the following Theorem 1.1: If G and H are closed subgroups of Sym(\(\Omega)\) and \(| G:H| <2^{\aleph_ 0}\) then H contains a pointwise stabilizer in G of some finite subset of \(\Omega\). The papers by \textit{J. D. Dixon}, \textit{P. M. Neumann} and \textit{S. Thomas} [Bull. Lond. Math. Soc. 18, 580-586 (1986; Zbl 0607.20003)] and \textit{D. M. Evans} [Bull. Lond. Math. Soc. 18, 587- 590 (1986; Zbl 0603.20041)] include theorems which are special cases of this result. A slightly weaker theorem has appeared in work in logic and been published by \textit{D. W. Kueker} [in Syntax Semantics Infinitary Languages, Lect. Notes Math. 72, 152-165 (1968; Zbl 0235.02018)] and by \textit{G. E. Reyes} [Ann. Math. Logic 1, 95-137 (1970; Zbl 0217.305)]. The theorem leads to a reformulation of an outstanding conjecture of H. D. Macpherson: If H is a subgroup of Sym(\(\Omega)\) such that H has only a finite number of orbits on \(\Omega\) k for each \(k\geq 0\), then the index of H in its closure is either 1 or \(2^{\aleph_ 0}\).
scientific article; zbMATH DE number 4033980