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Representations of Some Groups and Galois Stability

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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 \(G\subset GL_n(E)\) 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 \(G\subset GL_n(E)\), and if all coefficients of matrices in \(G\) are algebraic integers. Some related results and conjectures are considered.

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Acknowledgments

The authors are grateful to the referees for useful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, UWI, Mona Campus, Kingston, Jamaica

    Dmitry Malinin

  2. Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, Skudai, Johor, Malaysia

    Nor Haniza Sarmin, Nor Muhainiah Mohd Ali, Zainab Yahya & Noor Asma’ Adny Mohd Adnan

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  1. Dmitry Malinin
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  2. Nor Haniza Sarmin
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  3. Nor Muhainiah Mohd Ali
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  4. Zainab Yahya
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  5. Noor Asma’ Adny Mohd Adnan
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Correspondence to Dmitry Malinin.

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Communicated by Kar ** Shum.

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Malinin, D., Sarmin, N.H., Mohd Ali, N.M. et al. Representations of Some Groups and Galois Stability. Bull. Malays. Math. Sci. Soc. 38, 827–840 (2015). https://doi.org/10.1007/s40840-014-0051-7

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  • DOI: https://doi.org/10.1007/s40840-014-0051-7

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