Powerfully solvable and powerfully simple groups

TL;DR

The paper introduces the concept of powerfully solvable groups, a class of powerful groups with a specific abelian series, and explores their properties including a Jordan-Hölder type theorem for powerfully simple groups.

Contribution

It defines powerfully solvable and powerfully simple groups, establishing their properties and a Jordan-Hölder type theorem, expanding the theory of powerful groups.

Findings

Powerfully solvable groups include powerfully nilpotent groups.

A natural notion of powerfully simple groups is introduced.

A Jordan-Hölder type theorem for these groups is proved.

Abstract

We introduce the notion of a powerfully solvable group. These are powerful groups possessing an abelian series of a special kind. These groups include in particular the class of powerfully nilpotent groups. We will also see that for a certain rich class of powerful groups we can naturally introduce the term powerfully simple group and prove a Jordan-H\"older type theorem that justifies the term.