Describing finite groups by short first-order sentences

TL;DR

This paper demonstrates that finite simple groups and all finite groups can be described by very short first-order sentences, with complexity bounds related to the logarithm of their size, leveraging deep group-theoretic results.

Contribution

It establishes new bounds on the complexity of first-order descriptions for classes of finite groups, notably finite simple groups and all finite groups, using advanced algebraic techniques.

Findings

Finite simple groups are $ ext{log}$-compressible.

All finite groups are $ ext{log}^3$-compressible.

Finite transitive permutation groups are $ ext{log}^3$-compressible.

Abstract

We say that a class of finite structures for a finite first-order signature is $r$-compressible if each structure $G$ in the class has a first-order description of size at most $O(r(|G|))$. We show that the class of finite simple groups is $\log$-compressible, and the class of all finite groups is $\log^3$-compressible. As a corollary we obtain that the class of all finite transitive permutation groups is $\log^3$-compressible. The result relies on the classification of finite simple groups, the bi-interpretability of the twisted Ree groups with finite difference fields, the existence of profinite presentations with few relators, and group cohomology. We also indicate why the results are close to optimal.