On minimal coverings of groups by proper normalizers

TL;DR

This paper investigates the conditions under which groups, especially p-groups, can be covered by proper normalizers, and identifies certain non-nilpotent groups with nilpotent normalizer coverings.

Contribution

It introduces the concept of normalizer coverings in groups and characterizes p-groups without such coverings, also identifying non-nilpotent groups with nilpotent normalizer coverings.

Findings

Some p-groups lack a normalizer covering.

Certain non-nilpotent groups have a nilpotent normalizer covering.

The study advances understanding of group coverings by normalizers.

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.