Normal 2-coverings in affine groups of small dimension

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.