Representations of some groups and galois stability

• Main Authors: Malinin, Dmitry A., Sarmin, Nor Haniza, Mohd. Ali, Nor Muhainiah, Yahya, Zainab, Mohd. Adnan, Noor Asma’ Adny
• Format: Article
• Published: Springer Science+Business Media Singapore Private Limited 2015
• Subjects: QA Mathematics
• Online Access: http://eprints.utm.my/id/eprint/54621/; http://dx.doi.org/10.1007/s40840-014-0051-7
• Institution: Universiti Teknologi Malaysia

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.