On groups with all subgroups subnormal or soluble of bounded derived length

TL;DR

This paper investigates locally graded groups where subgroups are either subnormal or soluble with bounded derived length, establishing their structure and classifying certain finite simple groups with metabelian proper subgroups.

Contribution

It characterizes the structure of locally graded groups with specific subgroup properties and classifies finite simple groups with all proper subgroups metabelian.

Findings

Locally (soluble-by-finite) groups with the property are either soluble or extensions of soluble groups by finite groups.

Finite non-abelian simple groups with all proper subgroups metabelian are classified.

Provides structural insights into groups with subnormal or bounded derived length subgroups.

Abstract

In this paper, we deal with locally graded groups whose subgroups are either subnormal or soluble of bounded derived length, say d. In particular, we prove that every locally (soluble-by-finite) group with this property is either soluble or an extension of a soluble group of derived length at most d by a finite group, which fits between a minimal simple group and its automorphism group. We also classify all the finite non-abelian simple groups whose proper subgroups are metabelian.