On power maps in algebraic groups.
Let \(K\) be a field of characteristic exponent \(p\). (So if \(p=1\) then \(K\) has characteristic zero and otherwise it has characteristic \(p\).) The goal of this short note is to prove: Let \(G\) be a (connected) semisimple algebraic group over \(K\) and \(n\) a positive integer which is prime to \(p\). Then the power map \(P_n\) on \(G\) defined by \(x\mapsto x^n\) is surjective if and only if \(n\) is prime to the bad primes for \(G\) and also to the central primes, those that divide the order of the center of \(G\). This removes the restriction to characteristic zero in theorem C of \textit{P. Chatterjee} [Math. Res. Lett. 9, No. 5-6, 741-756 (2002; Zbl 1025.20034)]. -- See also the preceding review Zbl 1045.20043.
scientific article; zbMATH DE number 2064718