Describing finite groups by short first-order sentences

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.