The unipotent groups are an important class of algebraic groups. We show that techniques used to compute with finitely generated nilpotent groups carry over to unipotent groups. We concentrate particularly on maximal unipotent subgroups of split reductive groups and show how this improves computation in the reductive group itself.

Suppose $r=p^b$, where $p$ is a prime. Let $V$ be an $n$-dimensional ${\rm GF}(r)$-space and $G$ a subgroup of ${\rm A\Gamma L}(V) \cong {\rm A\Gamma L}(n,r)$ containing all translations and acting primitively on the set of vectors in $V$. Denote by $G_0$ the stabilizer in $G$ of the zero vector, so that $G_0 \le {\rm \Gamma L}(V) \cong {\rm \Gamma L}(n,r)$ and $G$ is the semidirect product of $V$ and $G_0$. Suppose that the generalized Fitting subgroup $F^*(G_0)$ of $G_0$ is an exceptional (twisted or untwisted, quasisimple) Chevalley group and that $\Gamma$ is a graph structure on $V$ on which $G$ acts primitively and distance transitively. The content of this paper is that then $G$ and $\Gamma$ are known. This result solves an open case in the outstanding problem of classifying all finite primitive distance-transitive groups. 2000 Mathematics Subject Classification: primary 20B25; secondary 05C25, 20Gxx, 05E30.

Borovik found a Lie primitive subgroup of E8(ℂ) isomorphic to (Alt5 × Sym6) : 2. In this note, we provide a short proof of existence and his result that the conjugacy class of this subgroup is the only one among those of non-local Lie primitive subgroups of finite dimensional simple complex Lie groups having a socle with more than one simple factor.