Representations of some groups and galois stability

Abstract

We study arithmetic problems for representations of finite groups over algebraic number fields and their orders under the ground field extensions. Let E/F be a Galois extension, and let G ? GL n (E) be a subgroup stable under the natural operation of the Galois group of E/F. A concept generalizing permutation modules is used to determine the structure of groups G and their realization fields. We also compare the possible realization fields of G in the cases if G ? GL n (E), and if all coeffi-cients of matrices in G are algebraic integers. Some related results and conjectures are considered.