A note on meta and para-$\mathfrak{Nil}$-Hamiltonian groups_v3

TL;DR

This paper investigates the conditions under which certain classes of groups, specifically meta and para-$ ext{ extit{Nil}}$-Hamiltonian groups, exhibit finiteness properties related to their subgroups and derived subgroups, focusing on soluble and non-nilpotent cases.

Contribution

It provides new insights into the finiteness conditions of meta and para-$ ext{ extit{Nil}}$-Hamiltonian groups with specific subgroup properties.

Findings

Finiteness of derived subgroups in meta-$ ext{ extit{Nil}}$-Hamiltonian groups with soluble subgroups.

Finiteness conditions for para-$ ext{ extit{Nil}}$-Hamiltonian groups containing non-nilpotent subgroups.

Characterization of group structures based on subgroup finiteness properties.

Abstract

Let $\mathfrak{Nil}$ be the class of nilpotent groups. This article explores the finiteness of meta and para-$\mathfrak{Nil}$-Hamiltonian groups or their derived subgroups when these groups contain a soluble subgroup of finite index or a non-nilpotent (or insoluble) subgroup of finite order respectively.