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Barely transitive locally nilpotent \(p\)-groups. (English) Zbl 0897.20028
The following notion was introduced by B. Hartley [Algebra Logika 13, 589-602 (1974; Zbl 0305.20019)]: A group \(G\) of permutations of an infinite set \(X\) is said to be barely transitive if \(G\) itself is transitive on \(X\) while every orbit of any proper subgroup of \(G\) is finite. Moreover, a group \(G\) is called a CC-group iff \(G/C_G(x^G)\) is Chernikov for every \(x\in G\).
By a theorem of B. Love, if \(G\) is locally finite and \(G'\neq G\) then \(G\) is a locally nilpotent \(p\)-group of Heineken-Mohamed type, but it is not known whether perfect barely transitive locally nilpotent \(p\)-groups exist.
In this paper it is shown that a barely transitive locally nilpotent \(p\)-group cannot be perfect if the stabilizer of a point is hypercentral and solvable. Two corollaries concerning locally nilpotent \(p\)-groups such that any proper subgroup is an FC-group or a CC-group are proved in addition.

MSC:
20F19 Generalizations of solvable and nilpotent groups
20B07 General theory for infinite permutation groups
20F50 Periodic groups; locally finite groups
20F24 FC-groups and their generalizations
20E25 Local properties of groups
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