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Barely transitive permutation groups. (English) Zbl 0694.20004
A group of permutations G of an infinite set \(\Omega\) is called a barely transitive group if G acts transitively on \(\Omega\) and every orbit of every proper subgroup is finite. These groups were introduced by B. Hartley and examples of barely transitive, locally finite groups with \(G\neq G'\) are given by him [in Proc. Camb. Philos. Soc. 74, 11-15 (1973; Zbl 0264.20031) and Algebra Logika 13, 589-602 (1974; Zbl 0305.20019)] and the structure of a locally finite barely transitive group G with \(G\neq G'\) is reasonably well-understood. In this paper the structure of locally finite barely transitive groups with \(G=G'\) is studied and the following theorems are proved: If G is a locally finite, locally p- solvable barely transitive group containing a non-trivial element of order p, then: 1) G is a p-group 2) Every proper normal subgroup is nilpotent of finite exponent (p is always a prime number). A locally finite barely transitive group is a countable group. But the existence of a locally finite barely transitive group with \(G=G'\) is still unclear. Theorem \(1.3'\). If G is a countable locally finite simple group containing a semisimple element, then G is not barely transitive. In particular if a locally finite simple group can be written as a union of finite simple groups, then G can not be a barely transitive permutation group. In general, the question whether a locally finite simple group can be simple remains unclear.
Reviewer: M.Kuzucuoğlu

MSC:
20B22 Multiply transitive infinite groups
20E25 Local properties of groups
20E32 Simple groups
20F50 Periodic groups; locally finite groups
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