On the global breadth of finite groups with nontrivial partitions

TL;DR

This paper investigates the structural properties of finite groups with nontrivial partitions, introduces a new class of groups based on a specific size condition, and demonstrates that many important simple groups belong to this class.

Contribution

It characterizes the global breadth of such groups, links it to local breadth and cyclic subgroup order, and shows that key simple groups are part of the new class al.

Findings

al contains projective special linear, general linear, and Suzuki groups.

al's global breadth is achieved via local breadth and maximal cyclic subgroup order.

Many simple groups are shown to belong to al.

Abstract

In a series of recent contributions on the notion of global breadth $\mathbf{B}(G)$ of a finite group $G$, it was interesting to observe the structural conditions arising from the classification of finite groups of $\mathbf{B}(G)=8$. This motivated the study of a new class of finite groups, namely $\mathcal{H}=\{G \ | \ G \ \mbox{satisfies the condition } \ |G| \le \mathbf{B}(G)(\mathbf{B}(G) + 1)\}$ and very little is known about $\mathcal{H}$. Here we focus on the groups with nontrivial partitions (according to the terminology of Baer, Kegel and Kontorovich), determining first that $\mathbf{B}(G)$ is achieved via the local breadth in connection with the order of maximal cyclic subgroups. Then we show that $\mathcal{H}$ contains projective special linear groups, projective general linear groups and Suzuki groups, supporting the conjecture that all finite groups with nontrivial partitions belong to $\mathcal{H}$. The presence of large families of simple groups in $\mathcal{H}$ is shown for the first time here.