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Groups whose non-normal subgroups are either nilpotent or minimal non-nilpotent

Nasrin Dastborhan, Hamid Mousavi

Abstract

Let $\mathfrak{Nil}$ be the class of nilpotent groups and $G$ be a group. We call $G$ a meta-$\mathfrak{Nil}$-Hamiltonian group if any of its non-$\mathfrak{Nil}$ subgroups is normal. Also, we call $G$ a para-$\mathfrak{Nil}$-Hamiltonian group if $G$ is a non-$\mathfrak{Nil}$ group and every non-normal subgroup of $G$ is either a $\mathfrak{Nil}$-group or a minimal non-$\mathfrak{Nil}$ group. In this paper we investigate the class of finitely generated meta-$\mathfrak{Nil}$-Hamiltonian and para-$\mathfrak{Nil}$-Hamiltonian groups.

Groups whose non-normal subgroups are either nilpotent or minimal non-nilpotent

Abstract

Let be the class of nilpotent groups and be a group. We call a meta--Hamiltonian group if any of its non- subgroups is normal. Also, we call a para--Hamiltonian group if is a non- group and every non-normal subgroup of is either a -group or a minimal non- group. In this paper we investigate the class of finitely generated meta--Hamiltonian and para--Hamiltonian groups.
Paper Structure (5 sections, 36 theorems, 7 equations)

This paper contains 5 sections, 36 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Primary lemmas
  3. The finite case
  4. Para-$\mathfrak{Nil}$-Hamiltonian and biminimal non-nilpotent groups
  5. Meta-$\mathfrak{Nil}$-Hamiltonian groups

Key Result

Theorem 2.1

Suzuki Let $q$ be a power of the prime $p$. Then, a subgroup of $\mathop{\mathrm{PSL}}\nolimits(2,q)$ is isomorphic to one of the following groups.

Theorems & Definitions (60)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 50 more