Weak second maximal subgroups in solvable groups

TL;DR

This paper studies the relationship between weak second maximal subgroups and second maximal subgroups in solvable groups, providing conditions for their coincidence and exploring implications for the structure of such groups.

Contribution

It offers a characterization of solvable groups where weak second maximal subgroups are identical to second maximal subgroups, termed WSM-groups, and examines their structural properties.

Findings

Characterization of WSM-groups in solvable groups

Condition for weak second maximal subgroups to coincide with second maximal subgroups

Non-vanishing elements in WSM-groups lie in the Fitting subgroup

Abstract

In this paper, we investigate the difference between weak second maximal subgroups and second maximal subgroups. A sufficient and necessary condition is also given to describe such class of groups whose weak second maximal subgroups coincide with its second maximal subgroups(called WSM-groups) under the solvable case. As an application, we show that every non-vanishing element of a solvable WSM-group lies in its Fitting subgroup.