Every finite group admits a just finite presentation

Let $L\subseteqΣ^*$ and fix a morphism $h:Σ^*\to M$ into a finite monoid. We study exact factorization and canonical presentation in the relative syntactic congruence $θ_{L,h}:=\equiv_L\cap\ker h$. We separate unique factorization from finite direct presentation. An exhaustively computer-checked $36$-element quotient has a unique exact prime factorization for every live non-unit class, yet its valid prime-return rules contain an infinite family, so unique factorization does not imply the finite relative presentation property (FRP), even for a finite quotient. We lift the same defect to a nonregular context-free language with an infinite relative quotient and finite prime spectrum. To isolate the obstruction, we introduce the finite-state relative presentation property (FSRP), in which canonical valid right-hand-side languages are represented by finite residual controllers, and prove $\mathrm{FRP}\subsetneq\mathrm{FSRP}$. We then introduce prime-target left-division determinism (PTLD), which implies unique exact factorization, tail exactness, tail determinism, and a quadratic bound on valid rules. A nonregular deterministic context-free example with a finite group observer satisfies PTLD while lying outside every fixed $(k,\ell)$-substitutable class. Finally, for fixed $h$ we give a strong positive-data learner for the canonical PTLD presentation with polynomial-time hypothesis updates and a finite characteristic sample, together with a limit reconstruction of the canonical FSRP controller from weakly behaviorally correct CFG-valued learners.

We introduce an extremal invariant associated with Chowla-type order conditions in finite groups. A nonempty subset $S$ of a finite group $G$ is called a Chowla set if every element of $S$ has order greater than $|S|$, and we write $C(G)$ for the maximum cardinality of such a set. We first show that $C(G)$ is determined by the distribution of element orders in $G$. For cyclic groups, we derive an exact divisor formula and characterize the integers $n$ for which $C(\mathbb{Z}/n\mathbb{Z})=\varphi(n)$. We prove that $\liminf_{n\to\infty}C(\mathbb{Z}/n\mathbb{Z})/\varphi(n)=1$, whereas $\limsup_{n\to\infty}C(\mathbb{Z}/n\mathbb{Z})/\varphi(n)=\infty$, and we determine the corresponding lower and upper limits under normalization by $n$. For finite abelian groups, we obtain an explicit formula in terms of the invariant-factor decomposition, together with a closed formula for finite abelian $p$-groups. We then develop a linear analogue for finite field extensions. A nonzero $K$-subspace $A$ of an extension $L/K$ is called a Chowla subspace if $[K(a):K]>\dim_K A$ for every nonzero $a\in A$. Since this condition depends on $\dim_K A$, it does not generally require every nonzero element of $A$ to generate $L$ over $K$. Nevertheless, when $L/K$ is finite and separable, we prove the exact formula $C(L/K)=[L:K]-d_{\max}(L/K)$, where $d_{\max}(L/K)$ is the largest degree over $K$ of a proper intermediate field. For finite fields, we give a direct proof in every degree using a normal-basis construction. This work was developed through an expert-guided human-AI collaboration. A reasoning-focused configuration of Co-Scientist was used to explore examples and potential proof strategies. The authors formulated the problem, independently verified and completed all arguments, and wrote the final proofs.

We present a Graph Neural Network (GNN) framework for the classification of finite groups according to their solvability. Using graph representations associated with finite groups, including Cayley graphs (CG), the proposed model is trained to distinguish solvable and non-solvable groups using structural graph information alone. The framework is evaluated on groups outside the training dataset in order to investigate the extent to which GNNs can learn algebraic properties arising in group theory. More broadly, the present work explores the relationship between algebraic structure and graph-based geometric representations of finite groups. The present study is intended as a proof-of-concept investigation of whether GNNs can learn algebraic properties of finite groups from graph-based representations