The proper definition and Wielandt-Hartley's theorem for submaximal $\mathfrak{X}$-subgroups

TL;DR

This paper investigates two definitions of submaximal $rak{X}$-subgroups within finite groups, establishing that Wielandt-Hartley's theorem applies to both, and explores applications of this strong theorem.

Contribution

It clarifies the proper definition of submaximal $rak{X}$-subgroups and proves Wielandt-Hartley's theorem holds for both definitions, enhancing understanding of subgroup maximality.

Findings

Wielandt-Hartley's theorem applies to both definitions of submaximal $rak{X}$-subgroups.

The two proposed definitions of submaximal $rak{X}$-subgroups are not equivalent.

Applications of the strong version of Wielandt-Hartley's theorem are provided.

Abstract

A nonempty class $\mathfrak{X}$ of finite groups is called complete if it is closed under taking subgroups, homomorphic images and extensions. We deal with a classical problem of determining $\mathfrak{X}$-maximal subgroups. We consider two definitions of submaximal $\mathfrak{X}$-subgroups suggested by Wielandt and discuss which one better suits our task. We prove that these definitions are not equivalent yet Wielandt-Hartley's theorem holds true for either definition of $\mathfrak{X}$-submaximality. We also give some applications of the strong version of Wielandt-Hartley's theorem.