Describing finite groups by short first-order sentences

Abstract

Abstract. We say that a class of finite structures for a finite firstorder signature is r-compressible for an unbounded function r : N → N + 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 logcompressible, 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 results rely 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 for finite groups, and group cohomology. We also indicate why the results are close to optimal.

Algorithmic search in group theory

“…The polynomials have non-negative integer coefficients. For example the description 8, 2, 3, 1 p ; 6, 7, 4, 2 q ; 15 with 2 degree 3 polynomials describes the integer (8x 3 + 2x 2 + 3x + 1) • (6x 3 + 7x 2 + 4x + 2)(15) = 83879080636024 which gets mapped to the pair of permutations (1,7,8,11,10,6,2,3,4,9,5) (1,2,6,5,7,3,4,10,11,9,8). Figure 2 shows results obtained by selecting 1,000,000 descriptions uniformly at random from the set of all descriptions with 7 degree 2 polynomials, ax 2 + bx + c, satisfying 1 ≤ a ≤ 20, 0 ≤ b, c ≤ 20, and with |v| ≤ 1000.…”

“…We require a few elementary results from the theory of Kolmogorov complexity. Since applications of Kolmogorov compexity to group theory are rare (we know of only [2,4,8]), we sketch proofs. For a more complete introduction to the theory, the reader is referred to [6] and [10].…”

“…The polynomials have non-negative integer coefficients. For example the description 8, 2, 3, 1 p ; 6, 7, 4, 2 q ; 15 with 2 degree 3 polynomials describes the integer (8x 3 + 2x 2 + 3x + 1) • (6x 3 + 7x 2 + 4x + 2)(15) = 83879080636024 which gets mapped to the pair of permutations (1,7,8,11,10,6,2,3,4,9,5) (1,2,6,5,7,3,4,10,11,9,8).…”

On the parallel complexity of Group Isomorphism via Weisfeiler-Leman

“…In the setting of graphs, the descriptive complexity has been extensively studied, with [Gro17] serving as a key reference in this area. There has been recent work relating first order logics and groups [NT17], as well as work examining the descriptive complexity of finite abelian groups [Gom10]. However, the work on the descriptive complexity of groups is scant compared to the algorithmic literature on GpI.…”

Model theory of finite and pseudofinite groups

“…4.1.4. Theorem 4.1.1 was recently used by Nies and Tent[51] to show that (1) finite simple groups are log-compressible, i.e., if G is a finite simple group, there is a first order sentence φ in the language L gp , with unique model G, such that φ has length O(log|G|), and more generally (2) for any finite group G there is such a sentence φ of length O((log|G|) 3 ).Remark 4.1.5. The model theory of any non-principal ultraproduct Π n∈N Alt n /U is undecidable.…”