Locally graded groups with all non-nilpotent subgroups permutable. II (Q6146655)

Recall that a group \(G\) is \textit{locally graded} if every non-trivial finitely generated subgroup of G has a non-trivial finite quotient. Recall also that a subgroup \(H\) of a group \(G\) is \textit{permutable} (or \textit{quasinormal}) in \(G\) if \(HK=KH\) for all subgroups \(K\) of \(G\). It is clear that this property is a generalization of normality. In a recent paper [\textit{S. Atlihan} et al., J. Algebra 632, 62--69 (2023; Zbl 1520.20057)] , the authors considered locally graded non-periodic groups in which every non-nilpotent subgroup is permutable, showing in particular that these groups are always soluble. The present paper deals with the periodic case, settling down the general case. The main result is the following. \textbf{Theorem}. Let \(G\) be a locally graded group and suppose every non-nilpotent subgroup of \(G\) is permutable. Then \(G\) is soluble.