an:01051760
Zbl 0897.20028
Asar, A. O.
Barely transitive locally nilpotent \(p\)-groups
EN
J. Lond. Math. Soc., II. Ser. 55, No. 2, 357-362 (1997).
00040417
1997
j
20F19 20B07 20F50 20F24 20E25
barely transitive permutation groups; locally finite groups; locally nilpotent \(p\)-groups; hypercentral subgroups; solvable subgroups; FC-groups; Chernikov groups; Heineken-Mohamed groups
The following notion was introduced by \textit{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.
W.Knapp (T??bingen)
Zbl 0305.20019