Jabara, Enrico Actions of Abelian groups on groups. (English) Zbl 1137.20022 J. Group Theory 10, No. 2, 185-194 (2007). Theorem 1: Let \(G\) be a group, and \(A\) a finitely generated Abelian subgroup of \(\operatorname{Aut}(G)\). If \(G\) is the union of finitely many \(A\)-orbits, then \(G\) is finite. Earlier, this type of result was obtained by P. M. Neumann and P. J. Rowley [Lond. Math. Soc. Lect. Note Ser. 252, 291-295 (1998; Zbl 0952.20022)] under the additional assumption that every non-trivial element of \(A\) acts without non-trivial fixed points. Theorem 2: Let \(G\) be a group, and \(H\) a virtually cyclic subgroup of \(G\). If \(G\) is the union of finitely many double cosets of \(H\), then either (a) \(H\) has finite index in \(G\), or (b) there is a characteristic subgroup \(R\) such that \(G/R\) and \(Z(R)\) are finite and \(R/Z(R)\) is an infinite simple group. Situations similar to Theorem 2 were considered earlier, for example, by G. A. Niblo [J. Algebra 220, No. 2, 512-518 (1999; Zbl 0944.20024)]. It is unknown whether case (b) in Theorem 2 is actually possible. The proofs of Theorems 1 and 2 involve a somewhat more general Theorem 3, which has a more technical statement, in terms of Abelian \(k\)-quasi-fixed-point-free groups of automorphisms. Theorem 3 gives an affirmative answer to a very special case of the following, still open, conjecture in the aforementioned paper by Neumann and Rowley: If a group \(G\) has an Abelian group of automorphisms with only finitely many orbits, then \(G\) is finite-by-Abelian-by-finite. Reviewer: Eugenii I. Khukhro (Preston) Cited in 2 Documents MSC: 20E36 Automorphisms of infinite groups 20E34 General structure theorems for groups 20F28 Automorphism groups of groups 20E07 Subgroup theorems; subgroup growth Keywords:Abelian groups of automorphisms; orbits; double cosets; finiteness conditions; fixed-point-free automorphism groups Citations:Zbl 0952.20022; Zbl 0944.20024 PDFBibTeX XMLCite \textit{E. Jabara}, J. Group Theory 10, No. 2, 185--194 (2007; Zbl 1137.20022) Full Text: DOI References: [1] DOI: 10.1006/jabr.1999.7935 · Zbl 0944.20024 · doi:10.1006/jabr.1999.7935 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.