Some remarks on regular subgroups of the affine group

Let V be a vector space over a field F of characteristic p 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 by Caranti, Dalla Volta, Sala, we show that the set {t-I|t\in T} is a subalgebra of End_F(F+V), which is nilpotent when V has finite dimension. This allows a rather systematic construction of abelian regular subgroups.