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Non-simplicity of locally finite barely transitive groups. (English) Zbl 0904.20031
Let \(\Omega\) be an infinite set. The authors consider locally finite, transitive subgroups \(G\) of the full symmetric group \(\text{Sym}(\Omega)\) with the property that every orbit of every proper subgroup of \(G\) is finite. Firstly, the authors prove that such a group \(G\) cannot be simple. Secondly, they prove that if such a \(G\) is also a finitary subgroup of \(\text{Sym}(\Omega)\), then \(G\) is a minimal non-FC, \(p\)-group for some prime \(p\). Thirdly, in the latter case, \(G\) cannot be generated by countably many elements of order \(p\). Finally, given such a \(G\), if the stabilizer of a point is Abelian, then \(G\) is a meta-Abelian \(p\)-group for some prime \(p\) with \(G/G'\) a Prüfer group and \(G'\) elementary Abelian.

MSC:
20F50 Periodic groups; locally finite groups
20B07 General theory for infinite permutation groups
20F24 FC-groups and their generalizations
20E07 Subgroup theorems; subgroup growth
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