Integral cohomology of classical groups over a finite field
Let \(G\) be one of the groups \(GL_n\), \(SL_n\), \(U_n\), \(SU_n\) or \(Sp\) over a finite field of characteristic \(p\). The author calculates the order of the integral Chern classes, obtained by Brauer lifting. The proof uses a description of the \(\text{mod }l\) cohomology of \(G\), for each prime \(l\) distinct from \(p\). For \(GL_n\) this description is done by \textit{D. Quillen} [Ann. Math., II. Ser. 96, 552-586 (1972; Zbl 0249.18022)]\ and for \(U_n\) and \(Sp\) by \textit{Z. Fiedorowicz} and \textit{S. Priddy} [Homology of classical groups over finite fields and their associated infinite loop spaces (Lect. Notes Math. 674, 1978; Zbl 0403.55010)]. The author computes this cohomology for the special groups \(SL_n\) and \(SU_n\).
scientific article; zbMATH DE number 166162
scientific article