Integer-valued representations of finite groups with the Galois action (Q1385938)

Let \(E/F\) be a normal extension of algebraic number fields with Galois group \(\Gamma= \text{Gal} (E/F)\). In this note without proofs, the author considers the problem of integer-valued realizations of finite subgroups \(G\) of a complete linear group \(GL_n(E)\) that are stable for a natural action of \(\Gamma\) upon the matrices of the group \(G\). He gets an existence criterion which is formulated for the integer-valued realization of an abelian group \(G\) of the type under consideration. Moreover, the author points out an error in theorem 1 of a previous article [\textit{D. A. Malinin}, Sov. Math., Dokl. 42, 773-776 (1991); translation from Dokl. Akad. Nauk SSSR 315, 299-302 (1990; Zbl 0781.20028)], but fills in most of the gap by the results of the paper under review.