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

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

top

## Abstract

top
Let $F$ be a field, $A$ be a vector space over $F$, $GL\left(F,A\right)$ be the group of all automorphisms of the vector space $A$. A subspace $B$ is called almost $G$-invariant, if ${dim}_{F}\left(B/{Core}_{G}\left(B\right)\right)$ is finite. In the current article, we begin the study of those subgroups $G$ of $GL\left(F,A\right)$ 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

top

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 - 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

top
1. Buckley J.T., Lennox J.C., Neumann B.H., Smith H., Wiegold J., 10.1017/S1446788700037289, J. Austral. Math. Soc. Ser. A 59 (1995), 384–398. Zbl0853.20023MR1355229DOI10.1017/S1446788700037289
2. Drozd Yu.A., Kirichenko V.V., Finite Dimensional Algebras, Vyshcha shkola, Kyiv, 1980. Zbl0816.16001MR0591671
3. Fuchs L., Infinite Abelian Groups, Vol. 1. Academic Press, New York, 1970. Zbl0338.20063MR0255673
4. Kargapolov M.I., Merzlyakov Yu.I., The Foundations of Group Theory, Nauka, Moscow, 1982. MR0677282
5. Kegel O.H., Wehrfritz B.A.F., Locally Finite Groups, North-Holland, Amsterdam, 1973. Zbl0259.20001MR0470081
6. Kurdachenko L., Otal J., Subbotin I., Artinian Modules Over Group Rings, Frontiers in Mathematics, Birkhäuser, Basel, 2007. Zbl1110.16001MR2270897
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
9. Robinson D.J.S., Finiteness Conditions and Generalized Soluble Groups, Part 1, Springer, New York, 1972. Zbl0243.20033MR0332989
10. Wehrfritz B.A.F., Infinite Linear Groups, Springer, Berlin, 1973. Zbl0261.20038MR0335656

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.