Infinite dimensional linear groups with many G — invariant subspaces

Let F be a field, A be a vector space over F , GL ( F, A ) be the group of all automorphisms of the vector space A . A subspace B of A is called nearly G-invariant , if dim F ( BFG/B ) is finite. A subspace B is called almost G-invariant , if dim F ( B / Core G ( B )) is finite. In the current article, we study linear groups G such that every subspace of A is either nearly G -invariant or almost G -invariant in the case when G is a soluble p -group where p = char F .