On non-periodic groups whose finitely generated subgroups are either permutable or pronormal. (English) Zbl 1264.20029

Kurdachenko, L. A.; Subbotin, I. Ya.; Ermolkevich, T. I.

Math. Bohem. 138, No. 1, 61-74 (2013).

A subgroup \(X\) of a group \(G\) is called pronormal if the subgroups \(X\) and \(X^g\) are conjugate in \(\langle X,X^g\rangle\) for each element \(g\) of \(G\). Moreover, the subgroup \(X\) is said to be permutable if \(XH=HX\) for all subgroups \(H\) of \(G\). In a previous paper [Asian-Eur. J. Math. 4, No. 3, 459-473 (2011; Zbl 1256.20038)] the authors described locally finite groups in which every finite subgroup is either pronormal or permutable. In the paper under review, it is proved that if \(G\) is a non-periodic generalized radical group in which every finitely generated subgroup is either pronormal or permutable, then all subgroups of \(G\) are permutable. Here, a group \(G\) is said to be generalized radical if it has an ascending (normal) series whose factors are either locally nilpotent or locally finite.

Reviewer: Francesco de Giovanni (Napoli)

MSC:

20E15	Chains and lattices of subgroups, subnormal subgroups
20E34	General structure theorems for groups
20E07	Subgroup theorems; subgroup growth
20F19	Generalizations of solvable and nilpotent groups
20F22	Other classes of groups defined by subgroup chains
20E25	Local properties of groups
20F14	Derived series, central series, and generalizations for groups

Keywords:

generalized radical groups; pronormal subgroups; permutable subgroups; finitely generated subgroups; abnormal subgroups

Citations:

Zbl 1256.20038