A survey of the number of supersolvable subgroups of finite groups

TL;DR

This survey explores criteria for finite group solvability based on counting supersolvable and non-supersolvable subgroups, including new examples and computational applications.

Contribution

It introduces new criteria for solvability related to subgroup counts and provides original examples and computational tools for analyzing finite groups.

Findings

New criteria for solvability based on subgroup counts

Examples of non-abelian supersolvable groups with two generators

Applications using GAP for subgroup analysis

Abstract

In this paper we survey a new criteria for solvability of finite groups in terms of number of supersolvable (also known as polycyclic) and non-supersolvable subgroups. In particular, we present original examples of supersolvable groups such as non-abelian groups with two cyclic generators. Additionally, we present some examples and applications in GAP system providing some description about recent criteria for solvability and properties of finite groups based on the number of supersolvable and non-supersolvable subgroups of a finite group $G$.