On finite groups whose Sylow subgroups are submodular

TL;DR

This paper investigates finite groups with submodular Sylow subgroups, establishing their structural properties and characterizing groups where all Sylow subgroups are submodular as Ore dispersive with strongly supersoluble biprimary subgroups.

Contribution

It introduces the concept of submodular Sylow subgroups and characterizes groups where all Sylow subgroups are submodular, linking them to Ore dispersive and strongly supersoluble biprimary subgroups.

Findings

A group with all Sylow subgroups submodular is Ore dispersive.

Such groups have all biprimary subgroups strongly supersoluble.

The paper provides necessary and sufficient conditions for these properties.

Abstract

A subgroup $H$ of a finite group $G$ is called submodular in $G$, if we can connect $H$ with $G$ by a chain of subgroups, each of which is modular (in the sense of Kurosh) in the next. If a group $G$ is supersoluble and every Sylow subgroup of $G$ is submodular in $G$, then $G$ is called strongly supersoluble. The properties of groups with submodular Sylow subgroups are obtained. In particular, we proved that in a group every Sylow subgroup is submodular if and only if the group is Ore dispersive and every its biprimary subgroup is strongly supersoluble.