Permutation groups on countable vector spaces over prime fields

TL;DR

This paper classifies all closed permutation groups acting on vectors of a countable vector space over a prime field of odd order that contain all automorphisms, showing their finiteness and correspondence to structures definable in the vector space.

Contribution

It provides a complete classification of such permutation groups and establishes their finiteness, linking them to first-order definable structures in the vector space.

Findings

Finite number of such permutation groups.

Correspondence with first-order definable structures.

Complete classification of automorphism groups.

Abstract

We describe all closed permutation groups which act on the set of vectors of a countable vector space $V$ over a prime field of odd order and which contain all automorphisms of $V$. In particular, we prove that their number is finite. These groups correspond, up to first-order interdefinability, precisely to all structures with a first-order definition in $V$.