Finite soluble groups with nilpotent wide subgroups

TL;DR

This paper characterizes finite soluble groups that lack wide subgroups and explores the structure of groups with nilpotent wide subgroups, revealing their quotient groups' properties.

Contribution

It provides a complete description of finite soluble groups without wide subgroups and analyzes the impact of nilpotent wide subgroups on group structure.

Findings

Finite soluble groups with no wide subgroups are fully characterized.

Groups with nilpotent wide subgroups have quotients with no wide subgroups.

Structural properties of groups related to wide subgroups are established.

Abstract

A subgroup of a finite group is wide if each prime divisor of the group order divides the subgroup order. We obtain the description of finite soluble groups with no wide subgroups. We also prove that a finite soluble group with nilpotent wide subgroups has the quotient group by its hypercenter with no wide subgroups.