On the solvability of a finite group by the sum of subgroup orders

TL;DR

This paper investigates conditions under which a finite group is solvable, showing that if the average order of its subgroups is below a certain threshold and certain subgroup conjugacy restrictions are met, then the group is solvable.

Contribution

It provides a partial solution to an open problem by establishing a solvability criterion based on subgroup order averages and conjugacy class restrictions.

Findings

If (G)<117/20, then G is solvable.

The result depends on restrictions on conjugacy classes of non-normal maximal subgroups.

Partial resolution of an open problem in group theory.

Abstract

Let $G$ be a finite group and $\sigma_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. Under some restrictions on the number of conjugacy classes of (non-normal) maximal subgroups of $G$, we prove that if $\sigma_1(G)<\frac{117}{20}\,$, then $G$ is solvable. This partially solves an open problem posed in \cite{9}.