Some remarks on regular subgroups of the affine group

‎Let $V$ be a vector space over a field $F$ of characteristic $p\geq ۰$ 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 $p>۰$‎, ‎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 ۶۹} (۲۰۰۶)‎, ‎۲۹۷--۳۰۸.]‎, ‎we show that the set $\left\{t-I\mid t\in T\right\}$ is a subalgebra‎ ‎of $End_F(F\oplus V)$‎, ‎which is nilpotent when $V$ has finite dimension‎. ‎This allows a rather systematic construction of abelian regular subgroups‎.

Keywords:

Affine group , Regular subgroups