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

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.

### 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

Zbl 1256.20038
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