Barely transitive permutation groups. (English) Zbl 0694.20004

Kuzucuoğlu, M.

Arch. Math. 55, No. 6, 521-532 (1990).

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

Keywords:

group of permutations; locally finite barely transitive groups; p-group; countable locally finite simple group; barely transitive permutation group

Citations:

Zbl 0264.20031; Zbl 0305.20019

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