BARELY TRANSITIVE LOCALLY NILPOTENT P-GROUPS

Abstract

B. Hartley in [4] introduced the class of barely transitive groups. By definition, a group of permutations G on an infinite set X is called barely transitive if G itself is transitive on X while every orbit of every proper subgroup of G is finite. If G is locally finite and G′≠G then the theorem of B. Love, proved in [4], shows that G is a locally nilpotent p-group of Heineken–Mohamed type. However, it is not known if perfect barely transitive locally nilpotent p-groups exist. Obviously this is a more general question than the corresponding one about perfect minimal non FC-groups (see below for definition). In this work it will be shown that a barely transitive locally nilpotent p-group cannot be perfect if the stabilizer of a point is hypercentral and solvable.