Finite Groups that are the union of at most 25 proper subgroups

TL;DR

This paper classifies finite groups with a union of at most 25 proper subgroups covering the entire group, introduces the concept of σ-elementary groups, and computes specific examples including the automorphism group of PSL(2,8).

Contribution

It provides a complete list of all σ-elementary groups with sum up to 25 and explores properties of these groups, including a specific calculation for Aut(PSL(2,8)).

Findings

Classified all σ-elementary groups with sum ≤ 25.

Identified σ(Aut(PSL(2,8))) as 29.

Introduced the concept of σ-elementary groups.

Abstract

For a finite group $G$ let $\sigma(G)$ (the "sum" of $G$) be the least number of proper subgroups of $G$ whose set-theoretical union is equal to $G$, and $\sigma(G)=\infty$ if $G$ is cyclic. We say that a group $G$ is $\sigma$-elementary if for every non-trivial normal subgroup $N$ of $G$ we have $\sigma(G)<\sigma(G/N)$. In this paper we produce the list of all the $\sigma$-elementary groups of sum up to 25. We also show that $\sigma(\Aut(PSL(2,8)))=29$.