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

Hamid Mousavi

TL;DR

The paper investigates finiteness properties of meta-$\mathfrak{Nil}$-Hamiltonian and para-$\mathfrak{Nil}$-Hamiltonian groups under the presence of a soluble subgroup of finite index or a finite-order non-nilpotent subgroup. It introduces and leverages the notion of $\mathcal{W}$-groups, proving that soluble-by-finite meta-$\mathfrak{Nil}$-Hamiltonian groups have finitely generated non-nilpotent subgroups with finite non-nilpotent images, and that many insoluble cases force finiteness or highly constrained structures. It further establishes that para-$\mathfrak{Nil}$-Hamiltonian groups with finite insoluble content are finite, and that when nilpotent subgroups have finite index, the group is polycyclic with a nilpotent $\mathrm{Fit}(G)$ of finite index; in certain finite quotients these groups reduce to well-known small simple groups like $A_5$ or $\mathrm{SL}(2,5)$. The results unify and extend prior work, offering a framework for solvability, polycyclicity, and finiteness classifications within meta- and para-$\mathfrak{Nil}$-Hamiltonian groups.

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.

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

TL;DR

The paper investigates finiteness properties of meta--Hamiltonian and para--Hamiltonian groups under the presence of a soluble subgroup of finite index or a finite-order non-nilpotent subgroup. It introduces and leverages the notion of -groups, proving that soluble-by-finite meta--Hamiltonian groups have finitely generated non-nilpotent subgroups with finite non-nilpotent images, and that many insoluble cases force finiteness or highly constrained structures. It further establishes that para--Hamiltonian groups with finite insoluble content are finite, and that when nilpotent subgroups have finite index, the group is polycyclic with a nilpotent of finite index; in certain finite quotients these groups reduce to well-known small simple groups like or . The results unify and extend prior work, offering a framework for solvability, polycyclicity, and finiteness classifications within meta- and para--Hamiltonian groups.

Abstract

Let be the class of nilpotent groups. This article explores the finiteness of meta and para--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.
Paper Structure (4 sections, 20 theorems, 9 equations)

This paper contains 4 sections, 20 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Primary and definition
  3. Meta-$\mathfrak{Nil}$-Hamiltonian group
  4. Para-$\mathfrak{Nil}$-Hamiltonian groups

Key Result

Theorem 2.1

AmFrGi Let the group $G = AB = AK = BK$ be the product of three nilpotent subgroups $A$, $B$, and $K$, where $K$ is normal in $G$. If $K$ is minimax, then $G$ is nilpotent.

Theorems & Definitions (34)

  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Lemma 2.7
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 24 more