Representations of some groups and Galois stability

Dmitry Malinin; Nor Haniza Sarmin; Nor Muhainiah Mohd Ali; Zainab Yahya; Noor Asma’ Adny Mohd Adnan

doi:10.1007/s40840-014-0051-7

Representations of some groups and Galois stability

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

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	827-840
Number of pages	14
Journal	Bulletin of the Malaysian Mathematical Sciences Society
Volume	38
Issue number	2
DOIs	https://doi.org/10.1007/s40840-014-0051-7
Publication status	Published - Apr 2015
Externally published	Yes

Keywords

Algebraic integers
Galois groups
Integral representations
Permutation modules and lattices
Realization fields

ASJC Scopus subject areas

General Mathematics

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10.1007/s40840-014-0051-7

Cite this

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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 ⊂ 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.",

keywords = "Algebraic integers, Galois groups, Integral representations, Permutation modules and lattices, Realization fields",

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AU - Malinin, Dmitry

AU - Sarmin, Nor Haniza

AU - Mohd Ali, Nor Muhainiah

AU - Yahya, Zainab

AU - Mohd Adnan, Noor Asma’ Adny

N1 - Publisher Copyright: © Malaysian Mathematical Sciences Society and Universiti Sains Malaysia 2014.

PY - 2015/4

Y1 - 2015/4

N2 - 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.

AB - 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.

KW - Algebraic integers

KW - Galois groups

KW - Integral representations

KW - Permutation modules and lattices

KW - Realization fields

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Representations of some groups and Galois stability

Abstract

Keywords

ASJC Scopus subject areas

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