**
Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 25, No. 1, pp. 17-27 (2009)
**

#
On finite linear groups stable under Galois operation

##
Ekaterina Khrebtova and Dmitry Malinin

Avango International, UAE and Belarusian National Technical University

**Abstract:** We consider a Galois extension $E/F$ of characteristic 0 and realization fields of finite abelian subgroups $G\subset GL_n(E)$ of a given exponent $t$. We assume that $G$ is stable under the natural operation of the Galois group of $E/F$. It is proven that under some reasonable restrictions for $n$ any $E$ can be a realization field of $G$, while if all coefficients of matrices in $G$ are algebraic integers there are only finitely many fields $E$ of realization having a given degree $d$ for prescribed integers $n$ and $t$ or prescribed $n$ and $d$. Some related results and conjectures are considered.

**Keywords:** integral representations, Galois group, algebraic integers, Galois algebras

**Classification (MSC2000):** 20C10; 11R33

**Full text of the article:**

[Previous Article] [Next Article] [Contents of this Number]

*
© 2009
FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition
*