Some remarks on regular subgroups of the affine group

Let $V$ be a vector space over a field $F$ of characteristic $pgeq 0$ and let $T$ be a regular subgroup of the affine group $AGL(V)$. In the finite dimensional case we show that, if $T$ is abelian or $p>0$, then $T$ is unipotent. For $T$ abelian, pushing forward some ideas used in [A. Caranti, F. Dalla Volta and M. Sala, Abelian regular subgroups of the affine group and radical rings, Publ. Math. Debrecen {bf 69} (2006), 297--308.], we show that the set $left{t-Imid tin Tright}$ is a subalgebra of $End_F(Foplus V)$, which is nilpotent when $V$ has finite dimension. This allows a rather systematic construction of abelian regular subgroups.