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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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