Rational matrix groups of a special type
Let the finite group G have a faithful irreducible complex character \(\chi\) such that \(\chi\) (1) is relatively prime to \(| G| /\chi (1)\). Let \(F=Q(\chi)\) and let \(n=[F:Q]\chi (1)\). Then G embeds in \(GL_ n(Q)\), with centralizer isomorphic to \(GL_ 1(F)\). The author shows that if \(\chi\) (1) and [F:Q] are not both even, then the normalizer of G in \(GL_ n(Q)\) splits over the centralizer. The proof uses cohomology and Clifford theory. The theorem was motivated by the special case of Chevalley groups of adjoint type, with \(\chi\) the Steinberg character.
scientific article; zbMATH DE number 3918630