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Infinite dimensional linear groups with a large family of $G$ -invariant subspaces

L. A. Kurdachenko; A. V. Sadovnichenko; I. Ya. Subbotin — 2010

Commentationes Mathematicae Universitatis Carolinae

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 ${dim}_{F} (B / {Core}_{G} (B))$ 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.

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