The small index property of automorphism groups of ab-initio generic structures

TL;DR

This paper proves that the automorphism group of certain countable generic structures has the small index property, meaning subgroups of small index are closely related to stabilizers of finite sets.

Contribution

It establishes the small index property for automorphism groups of ab-initio generic structures derived from rational pre-dimension functions.

Findings

Automorphism groups have the small index property.

Subgroups of index less than continuum are stabilizers of finite sets.

Supports the understanding of symmetry groups in generic structures.

Abstract

Suppose $M$ is a countable ab-initio (uncollapsed) generic structure which is obtained from a pre-dimension function with rational coefficients. We show that if $H$ is a subgroup of $\mbox{Aut}\left(M\right)$ with $\left[\mbox{Aut}\left(M\right):H\right]<2^{\aleph_{0}}$, then there exists a finite set $A\subseteq M$ such that $\mbox{Aut}_{A}\left(M\right)\subseteq H$. This shows that $\mbox{Aut}\left(M\right)$ has the small index property.