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

TL;DR

This paper classifies all closed permutation groups containing the infinite linear group over the two-element field and explores their associated first-order definable reducts in an infinite-dimensional vector space.

Contribution

It provides a complete classification of four closed subgroups between the infinite linear group and the symmetric group, and analyzes their reducts and groups containing the infinite symplectic group.

Findings

Exactly four such closed subgroups are identified.

The reducts of the infinite-dimensional vector space are characterized.

Groups containing the infinite symplectic group are classified.

Abstract

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