Locally graded groups with all non-nilpotent subgroups permutable, II

TL;DR

This paper proves that a locally graded group in which all non-nilpotent subgroups are permutable must be soluble, extending understanding of the structure of such groups.

Contribution

It establishes the solubility of locally graded groups with permutable non-nilpotent subgroups, generalizing previous results and focusing on periodic cases.

Findings

G is soluble under given conditions

Extension of previous results on permutability and solubility

Applicable to periodic locally graded groups

Abstract

Let $G$ be a locally graded group and suppose that every non-nilpotent subgroup of $G$ is permutable. We prove that $G$ is soluble. (In light of previous results of the authors, it suffices to prove that $G$ is soluble if it is periodic.