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.