Geodesic
Infinite dimensional linear groups with a large family of $G$-invariant subspaces
Commentationes Mathematicae Universitatis Carolinae, Tome 51 (2010) no. 4, pp. 551-558 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 15A03, 20F16, 20F29
Keywords: vector space; linear groups; periodic groups; soluble groups; invariant subspaces 
@article{CMUC_2010_51_4_a0,
     author = {Kurdachenko, L. A. and Sadovnichenko, A. V. and Subbotin, I. Ya.},
     title = {Infinite dimensional linear groups with a large family of $G$-invariant subspaces},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {551--558},
     year = {2010},
     volume = {51},
     number = {4},
     mrnumber = {2858259},
     zbl = {1224.15002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2010_51_4_a0/}
}
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Kurdachenko, L. A.; Sadovnichenko, A. V.; Subbotin, I. Ya. Infinite dimensional linear groups with a large family of $G$-invariant subspaces. Commentationes Mathematicae Universitatis Carolinae, Tome 51 (2010) no. 4, pp. 551-558. http://geodesic.mathdoc.fr/item/CMUC_2010_51_4_a0/