## Infinite dimensional linear groups with a large family of $$G$$-invariant subspaces.(English)Zbl 1224.15002

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.

### MSC:

 15A03 Vector spaces, linear dependence, rank, lineability 20F16 Solvable groups, supersolvable groups 20F29 Representations of groups as automorphism groups of algebraic systems
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