### Infinite dimensional linear groups with a large family of $G$-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 $B$ is called almost $G$-invariant, if ${dim}_{F}(B/{Core}_{G}\left(B\right))$ is finite. In the current article, we begin the study of those subgroups $G$ of $GL(F,A)$ 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.