Covering by Centralizers

TL;DR

This paper studies how centralizers of elements can cover finite groups, showing maximal and minimal centralizers form covers, and explores properties of nonabelian p-groups regarding their centralizers.

Contribution

It establishes that maximal and minimal centralizers form covers of finite groups and analyzes the number of nontrivial centralizers in nonabelian p-groups.

Findings

Maximal centralizers form a cover of the group.

Minimal centralizers form a cover of the group.

Number of nontrivial centralizers in nonabelian p-groups is congruent to 1 modulo p.

Abstract

In this paper, we consider covers of finite groups by centralizers of elements. We show that the set of centralizers that are maximal under the partial ordering form a cover of the group. We also show that the set of centralizers that are minimal under the partial ordering form a cover of the group. We show for $F$-groups that are nonabelian $p$-groups that the number of distinct nontrivial centralizers is congruent to $1$ modulo $p$.