Unipotent subgroups of reductive groups in characteristic \(p>0\) (Q1847949)

Let \(k\) be a field of positive characteristic \(p\), and \(G/k\) a reductive connected algebraic group. The author proves two conjectures of J. Tits on the unipotent elements of semisimple algebraic groups. The first conjecture says that if \([k:k^p]\leq p\), and \(G/k\) is semisimple and simply connected, then every unipotent subgroup of \(G(k)\) is \(k\)-embeddable into the unipotent radical of a \(k\)-parabolic subgroup of \(G\). The second one says that if \(G/k\) is split and almost simple, then every \(k\)-anisotropic automorphism of order \(p\) normalizes a maximal \(k\)-split torus of \(G\).