Finite groups which have maximal covers

TL;DR

This paper investigates finite groups with maximal irredundant subgroup covers, proving that those with a cover size close to the group order are solvable and providing a classification for these cases.

Contribution

It establishes solvability and classifies finite groups with maximal covers where the cover size is within five of the group order.

Findings

Finite groups with $oxed{ ext{maximal covers close to }|G|}$ are solvable.

Complete classification of groups with $oxed{ ext{cover size } |G|-t, t extless 6}$.

Identification of structural properties of these groups.

Abstract

Let $\lambda(G)$ be the maximum number of subgroups in an irredundant covering of a finite group $G$. We prove that the finite groups with $\lambda(G)=|G|-t$, where $t\leq 5$, are solvable, and classify such groups.