Covering by Centralizers

TL;DR

The paper investigates how finite groups can be covered by centralizers of noncentral elements and by centers of those centralizers, establishing that maximal and minimal centralizers (and their centers) form covers and that these structures connect to a centralizer graph whose dominating sets correspond to covers. It provides a graph-theoretic framework linking covers to domination in the centralizer graph and characterizes F- and CA-groups through partitions and maximal/minimality properties of centers and centralizers. A key result is that in nonabelian p-groups that are F-groups, the count of distinct noncentral centralizers is congruent to 1 modulo p. The work advances understanding of how centralizers organize group covers and offers a bridge between group-theoretic and graph-theoretic perspectives on covers.

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 $p$.

Figures (1)

• Figure 1: A Hasse diagram showing the centralizers and centers of $S_4$.

Theorems & Definitions (44)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Lemma 2.1
• Lemma 2.2
• Corollary 2.3