On some linear groups, having a big family of \(G\)-invariant subspaces

Abstract

Let \(F\) be a field, \(A\) a vector space over \(F\), \(GL(F, A)\) be the group of all automorphisms of the vector space \(A\). If \(B\) is a subspace of \(A\), then denote by \(BFG\) the \(G\)-invariant subspace, generated by \(B\). A subspace \(B\) is called nearly \(G\)-invariant, if \(dim_F(BFG/B)\) is finite. In this paper we described the situation when every subspace of \(A\) is nearly \(G\)-invariant.