Normal 2-coverings in affine groups of small dimension

TL;DR

This paper investigates finite groups with a normal 2-covering, focusing on the affine case, which is crucial for understanding the broader classification of such groups.

Contribution

It initiates a preliminary study of affine groups with normal 2-coverings, a key step in classifying all finite groups with this property.

Findings

Identifies the importance of affine groups in the classification of groups with normal 2-coverings.

Provides initial insights into the structure of affine groups with such coverings.

Sets the stage for further detailed analysis of affine cases.

Abstract

A finite group $G$ admits a normal $2$-covering if there exist two proper subgroups $H$ and $K$ with $G=\bigcup_{g\in G}H^g\cup\bigcup_{g\in G}K^g$. For determining inductively the finite groups admitting a normal $2$-covering, it is important to determine all finite groups $G$ possessing a normal $2$-covering, where no proper quotient of $G$ admits such a covering. Using terminology arising from the O'Nan-Scott theorem, Garonzi and Lucchini have shown that these groups fall into four natural classes: product action, almost simple, affine and diagonal. In this paper, we start a preliminary investigation of the affine case.