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Actions of Abelian groups on groups. (English) Zbl 1137.20022

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.

MSC:

20E36 Automorphisms of infinite groups
20E34 General structure theorems for groups
20F28 Automorphism groups of groups
20E07 Subgroup theorems; subgroup growth
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[1] DOI: 10.1006/jabr.1999.7935 · Zbl 0944.20024 · doi:10.1006/jabr.1999.7935
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