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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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