On the cohomology of finite groups of Lie type (Q1203486)

Let \(q\) be the number of elements of the field \(F_ q\) of characteristic \(p\). The author proves the following theorem: Let \(G\) be a connective reductive \(F_ q\)-group scheme and let \(l\) be a prime different from \(p\). Then the following are equivalent: (i) The inclusion \(G(F_ q) \to G(F_{q^ n})\) induces an \(H^*(\;;\mathbb{Z}/l)\)-isomorphism. (ii) The groups \(G(F_ q)\) and \(G(F_{q^ n})\) have isomorphic \(l\)-Sylow subgroups. This result extends to central quotients of \(G(F_{q^ n})\) and for Suzuki and Ree groups.