Infinite dimensional linear groups with a large family of G -invariant subspaces

L. A. Kurdachenko; A. V. Sadovnichenko; I. Ya. Subbotin

Commentationes Mathematicae Universitatis Carolinae (2010)

  • Volume: 51, Issue: 4, page 551-558
  • ISSN: 0010-2628

Abstract

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Let F be a field, A be a vector space over F , GL ( F , A ) be the group of all automorphisms of the vector space A . A subspace B is called almost G -invariant, if dim F ( B / Core G ( B ) ) is finite. In the current article, we begin the study of those subgroups G of GL ( F , A ) for which every subspace of A is almost G -invariant. More precisely, we consider the case when G is a periodic group. We prove that in this case A includes a G -invariant subspace B of finite codimension whose subspaces are G -invariant.

How to cite

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Kurdachenko, L. A., Sadovnichenko, A. V., and Subbotin, I. Ya.. "Infinite dimensional linear groups with a large family of $G$-invariant subspaces." Commentationes Mathematicae Universitatis Carolinae 51.4 (2010): 551-558. <http://eudml.org/doc/246876>.

@article{Kurdachenko2010,
abstract = {Let $F$ be a field, $A$ be a vector space over $F$, $\operatorname\{GL\}(F,A)$ be the group of all automorphisms of the vector space $A$. A subspace $B$ is called almost $G$-invariant, if $\dim _\{F\}(B/\operatorname\{Core\}_\{G\}(B))$ is finite. In the current article, we begin the study of those subgroups $G$ of $\operatorname\{GL\}(F,A)$ for which every subspace of $A$ is almost $G$-invariant. More precisely, we consider the case when $G$ is a periodic group. We prove that in this case $A$ includes a $G$-invariant subspace $B$ of finite codimension whose subspaces are $G$-invariant.},
author = {Kurdachenko, L. A., Sadovnichenko, A. V., Subbotin, I. Ya.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {vector space; linear groups; periodic groups; soluble groups; invariant subspaces; vector space; linear group; periodic group; soluble group; invariant subspace; automorphisms},
language = {eng},
number = {4},
pages = {551-558},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Infinite dimensional linear groups with a large family of $G$-invariant subspaces},
url = {http://eudml.org/doc/246876},
volume = {51},
year = {2010},
}

TY - JOUR
AU - Kurdachenko, L. A.
AU - Sadovnichenko, A. V.
AU - Subbotin, I. Ya.
TI - Infinite dimensional linear groups with a large family of $G$-invariant subspaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2010
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 51
IS - 4
SP - 551
EP - 558
AB - Let $F$ be a field, $A$ be a vector space over $F$, $\operatorname{GL}(F,A)$ be the group of all automorphisms of the vector space $A$. A subspace $B$ is called almost $G$-invariant, if $\dim _{F}(B/\operatorname{Core}_{G}(B))$ is finite. In the current article, we begin the study of those subgroups $G$ of $\operatorname{GL}(F,A)$ for which every subspace of $A$ is almost $G$-invariant. More precisely, we consider the case when $G$ is a periodic group. We prove that in this case $A$ includes a $G$-invariant subspace $B$ of finite codimension whose subspaces are $G$-invariant.
LA - eng
KW - vector space; linear groups; periodic groups; soluble groups; invariant subspaces; vector space; linear group; periodic group; soluble group; invariant subspace; automorphisms
UR - http://eudml.org/doc/246876
ER -

References

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  7. Kurdachenko L.A., Sadovnichenko A.V., Subbotin I.Ya., 10.2478/s11533-009-0007-6, Cent. Eur. J. Math. 7 (2009), no. 2, 176–185. MR2506958DOI10.2478/s11533-009-0007-6
  8. Pierce R.S., Associative Algebras, Springer, Berlin, 1982. Zbl0671.16001MR0674652
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