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On minimal coverings of groups by proper normalizers

M. Amiri, S. Haji, S. M. Jafarian Amiri

Abstract

For a group $G$, a {\it normalizer covering} of $G$ is a finite set of proper normalizers of some subgroups of $G$ whose union is $G$. We study $p$-groups ($p$ a prime) without a normalizer covering. As an application, we determine some non-nilpotent groups having a nilpotent normalizer covering.

On minimal coverings of groups by proper normalizers

Abstract

For a group , a {\it normalizer covering} of is a finite set of proper normalizers of some subgroups of whose union is . We study -groups ( a prime) without a normalizer covering. As an application, we determine some non-nilpotent groups having a nilpotent normalizer covering.

Paper Structure

This paper contains 4 sections, 19 theorems, 11 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries results
  3. normalizer coverings of nilpotent groups
  4. An application

Key Result

Lemma 2.1

Let $G$ be a group. Then the following statements are equivalent: (i)$\sigma_{ \mathfrak{n}} (G)=\infty$. (ii)there exists $x\in G$ whenever $x\in N_G(H)$ for some subgroup $H$ of $G$, then $H\unlhd G$.

Theorems & Definitions (40)

  • Remark 1.1
  • Conjecture 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 30 more