Non-nilpotent subgroups of locally graded groups

TL;DR

This paper investigates the structure of locally graded groups with finitely many non-nilpotent subgroups, establishing bounds on their soluble and derived lengths based on subgroup properties.

Contribution

It provides new bounds on the soluble and derived lengths of locally graded groups with finitely many non-nilpotent subgroups, linking subgroup structure to group solvability.

Findings

Bound on soluble length: at most [log2(n)] + m + 3

Bound on derived length: at most [log2(n)] + m + 1

Characterization of groups with finitely many non-(nilpotent of class ≤ n) subgroups

Abstract

In this paper, we show that a locally graded group with a finite number m of non-(nilpotent of class at most n) subgroups is (soluble of class at most [log2(n)] + m + 3)-by-(finite of order $\le$ m!). Also we show that the derived length of a soluble group with a finite number m of non-(nilpotent of class at most n) subgroups, is at most [log2(n)] + m + 1.