Maximal covers of finite groups

TL;DR

This paper investigates the maximum number of subgroups in irredundant coverings of finite groups, establishing conditions under which such groups are supersolvable or solvable, and describing the structure of groups with specific maximum counts.

Contribution

It proves that groups with at most 6 subgroups in an irredundant covering are supersolvable and characterizes groups with exactly 6, also showing groups with up to 30 are solvable.

Findings

Groups with λ(G) ≤ 6 are supersolvable.

Structural description of groups with λ(G) = 6.

Groups with λ(G) ≤ 30 are solvable.

Abstract

Let $\lambda(G)$ be the maximum number of subgroups in an irredundant covering of the finite group $G$. We prove that if $G$ is a group with $\lambda(G) \leqslant 6$, then $G$ is supersolvable. We also describe the structure of the groups $G$ with $\lambda(G)=6$. Moreover, we show that if $G$ is a group with $\lambda(G) \leqslant 30$, then $G$ is solvable.