Permutation groups containing infinite linear groups and reducts of infinite dimensional linear spaces over the two element field

Let $\mathbb{F}_2^ω$ denote the countably infinite dimensional vector space over the two element field and $\operatorname{GL}(ω, 2)$ its automorphism group. Moreover, let $\operatorname{Sym}(\mathbb{F}_2^ω)$ denote the symmetric group acting on the elements of $\mathbb{F}_2^ω$. It is shown that there are exactly four closed subgroups, $G$, such that $\operatorname{GL}(ω, 2)\leq G\leq \operatorname{Sym}(\mathbb{F}_2^ω)$. As $\mathbb{F}_2^ω$ is an $ω$-categorical (and homogeneous) structure, these groups correspond to the first order definable reducts of $\mathbb{F}_2^ω$. These reducts are also analyzed. In the last section the closed groups containing the infinite symplectic group $\operatorname{Sp}(ω, 2)$ are classified.

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