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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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