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Description of barely transitive groups with soluble point stabilizer. (English) Zbl 1181.20002
Let \(G\) be a group with a faithful transitive permutation representation on an infinite set such that every orbit of every proper subgroup of \(G\) is finite. (Such a group is said to be barely transitive.) The authors describe such groups with an Abelian-by-finite (respectively nilpotent-by-finite, respectively soluble-by-finite) point stabilizer. If \(G\) is infinitely generated and a point stabilizer is a permutable subgroup, they show that \(G\) is locally finite. B. Hartley [Algebra Logika 13, 589-602 (1974; Zbl 0305.20019); translation in Algebra Logic 13(1974), 334-340 (1975)] asked 35 years ago whether there exist torsion-free barely transitive groups. It seems that this is still unanswered, but the authors at least show that there is no torsion-free barely transitive group with a nilpotent point stabilizer.

MSC:
20B07 General theory for infinite permutation groups
20F50 Periodic groups; locally finite groups
20E25 Local properties of groups
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