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Keywords:
vector space; linear groups; periodic groups; soluble groups; invariant subspaces
vector space; linear groups; periodic groups; soluble groups; invariant subspaces
Summary:
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.
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