On the surjectivity of the power maps of semisimple algebraic groups. (Q1429368)

The author formulates and proves the analogue in positive characteristic of theorem C of \textit{P. Chatterjee} [Math. Res. Lett. 9, No. 5-6, 741-756 (2002; Zbl 1025.20034)]. This had been done a little earlier by \textit{R. Steinberg} [Math. Res. Lett. 10, No. 5-6, 621-624 (2003; see the following review Zbl 1045.20044)], but the proofs differ. Next he applies the result to tell when a connected semisimple algebraic group is `exponential' in the sense that every element is contained in a connected Abelian algebraic subgroup. Say the characteristic of the ground field is \(p>0\). Then the following are equivalent: 1.~\(G\) is exponential. 2.~The map \(x\mapsto x^n\) is surjective for all positive integers \(n\) prime to \(p\) and \(p\) is a good prime for \(G\). 3.~\(G\) is isogenous to a product of groups of type \(A_\ell\), \(\ell\geq 1\), and the reduced center of \(G\) is trivial.