Finite Groups

Symmetries play a central role in reducing the complexity of reinforcement learning problems, yet most existing approaches rely on fixed group actions or predefined state abstractions. Classical reinforcement learning algorithms typically assume a globally structured Markov decision process with uniformly applicable actions and transitions, an assumption that limits their ability to exploit modularity and local, context-dependent regularities present in many realistic environments. We propose a reinforcement learning framework using groupoids to capture local, state-dependent symmetries and support the dy- namic discovery of equivalence structures during interaction. The agent maintains orbit representatives together with transporters that map raw states to canonical forms, enabling learning and decision-making to be performed in a symmetry-reduced space while preserving local distinctions. Empirical results demonstrate that the proposed groupoid-based approach improves sample efficiency and convergence in dense and large-scale environments exhibiting strong partial symmetries, yielding substantial performance gains over standard Q-learning. These findings show that dynamically exploiting local symmetry provides a practical and mathematically principled route to scalable and generalisable reinforcement learning.

Smoothness of a function on the real line is reflected in the decay of its Fourier transform, which suggests that smoothness of a function in $L^2(G)$ for a group $G$ should mean concentration of the Fourier coefficients at low frequency. Such a reading presupposes an ordering of the irreducible representations of $G$, but for non-abelian $G$, no ordering is canonical. Given a symmetric generating set $S$, the Laplacian of the associated Cayley graph is block diagonal over the dual, and we order the irreps by the mean of the eigenvalues in each block. This produces an ordering function $ω:\widehat{G}\to\mathbb{R}$ that depends only on the pair $(G,S)$. This function is bounded between zero and two, vanishing only at the trivial representation and achieving the upper bound exactly when the Cayley graph is bipartite. We then ask how much freedom the construction has. Within the class of operators satisfying natural axioms, the induced orderings are exactly the real functions on the dual vanishing at the trivial representation and agreeing on conjugate pairs, and the orderings coming from inversion orbits of conjugacy classes form a basis for them. We cut the freedom down further by requiring two additional inputs: nonnegativity of the class weights and a declaration of which group elements count as uniform incremental changes, which pins the operator to the Cayley-Laplacian up to positive scale. We observe that the construction persists for compact groups even though the Cayley graph does not, and we extend the theory to finite sets carrying a transitive group action, where the acting group selects which frequencies exist and the generating set orders them. The answer to the title question is therefore that smoothness is a property of a function together with a choice of group and generating set, not of the function alone.

Reasoning shortcuts are solutions of a neurosymbolic system's rules that produce correct predictions through unintended concepts. A recent framework of Takemura, Inoue, and Nishino analyzes them through an automorphism group of value relabelings and asks, as its central open question, when rules pin concepts down. We first show that the framework's key definition, one shared permutation applied at every position, does not apply as stated to any of the four heterogeneous benchmarks it was evaluated on, and that the most direct embedding, padding domains to a common size, produces confident false pathology: 90.91% of solution pairs reported unexplained on CLE4EVR, where every well-defined member of the hierarchy we introduce reports 0%, and the padded verdict's content rotates with configuration-file ordering. Re-measuring eleven rule families under fifteen pre-specified predictions (thirteen confirmed), unexplained-pair rates span 0% to 99.9999% and track provable structure: six theorems give sufficient conditions for transitivity and its failure, including a Free Slot Lemma certifying Kandinsky's pathology from syntax alone. For circuit-given rules, deciding symmetry-inertness of a coordinate is coNP-complete; nontrivial-automorphism existence is coNP-hard under randomized reductions, lies in $Σ_2^p$, is not $Σ_2^p$-complete unless PH collapses, and on monotone circuits is coNP-complete outright. In the Boolean case transitivity is classified exactly: automorphisms explain everything iff the solution set is an affine coset. Weakly supervised models place all 94 observed shortcuts at the one level the componentwise theory flags and none at the 48 it certifies transitive; twelve typed-ambiguous levels produce none, separating what symmetry permits from what optimization selects, and a dual-head control replicates the geography. All numbers trace to released artifacts.

A group is an E-group if every element commutes with each of its endomorphic images. Caranti asked whether a finite E-group can have nilpotency class three. We prove that the $3$-group of order $3^{84}$ introduced by Abdollahi, Faghihi, and Mohammadi Hassanabadi, and later shown by Abdollahi, Faghihi, Linton, and O'Brien to have the corresponding automorphism property, is an E-group. Let $P$ denote this group and put $V=P/Φ(P)\cong \mathbb{F}_3^9$. The nine power relations of $P$ determine a linear map $q:V\longrightarrowΛ^2 V$. We prove that $q$ has no nonzero proper subspace $U$ satisfying $q(U)\subseteqΛ^2 U$. Since the image induced by any endomorphism of $P$ on $V$ has precisely this closure property, every endomorphism acts on $V$ either invertibly or trivially. The invertible case is the known A-group case. In the trivial case the image first lies in $Φ(P)=P'$, and the power relations then force it into $Ω_1(P')=Z(P)$. Thus every element commutes with every endomorphic image. The tensor rigidity is reduced to an exact finite calculation on the $9841$ points of $\mathrm{PG}(8,3)$.

A gradient-based GNN explainer given a molecule with two chemically equivalent nitro groups assigns them attribution scores that are equal to the last bit. It cannot do otherwise: message passing is exactly permutation equivariant, so any automorphism of the input leaves every attribution invariant. Yet the standard report, the top-k edges, names one of the two, and which one is settled by the order of an array. We show this is a structural obstruction rather than an implementation slip. When no minimal valid explanation is fixed by the input's automorphism group, no rule can be single-valued, minimal and symmetry-respecting at once. For the exact-k reports used in practice we give a parameter-free criterion, mechanised in Lean 4 with no axiom dependencies, that decides from the graph alone whether every score-optimal report of that size must split an orbit. Across 21298 instance-budget decisions the criterion agrees with a mechanical model-equivalence check without exception, and no severing case we found admitted a neutral alternative. The obstruction is common. Nontrivial automorphisms occur in 93.4% of Mutagenicity, the dataset the seminal explainability papers use, so the measure-zero dismissal of symmetric inputs, sound on the continuous domains it was made for, collapses here. At the sparsity budget those papers report, 24.0% of molecules with two interchangeable nitro groups (6 of 25) surface exactly one of them, every one arbitrary under mechanical verification. A model's blindness also manufactures symmetry: every MUTAG molecule contains atoms chemistry separates and the network provably cannot, and a matched control shows the resolution is set by what the model reads rather than how it is parameterised. Reporting orbits removes the arbitrariness at 0.11 ms and 0.43 extra edges per graph.

The Word Problem has been a subject of intensive mathematical study for over a century, initially driving advances in combinatorial group theory and more recently emerging as a foundational hardness assumption in post-quantum cryptography (PQC). While generally undecidable, several families of infinite non-abelian groups exhibit solvable or algorithmically fast word problems, making them attractive platforms for cryptographic design. This paper introduces WPNet, a novel Graph Neural Network architecture capable of solving the Word Problem heuristically, which is demonstrated on the Baumslag-Solitar group $BS(1,2)$ and on an Artin group. By mapping unreduced words to dynamic graph structures, the model learns to cluster algebraically equivalent elements in a continuous embedding space, effectively identifying the geodesic representative of a word without executing discrete reduction steps. As an application, a model variant is developed that can predict the geodesic length of an unreduced word in both groups. To demonstrate the cryptographic severity of this structural leakage, WPNet is successfully deployed against the Wagner-Magyarik public-key cryptosystem.

In Question~3.1 of his 1995 paper on depth and transfer, Carlson asked whether the depth of a finite-group cohomology ring is always realized by the dimension of one of its associated primes. We give a negative answer. Let $$G=\SG{128}{859},\qquad k=\kbar.$$ An exact presentation certificate proves that $\depth H^*(G;k)=2$. Okuyama's associated-prime theorem would convert an associated prime of dimension two into a rank-two elementary abelian subgroup $E\leq G$ satisfying $\depth H^*(C_G(E);k)=2$. We enumerate all $75$ rank-two elementary abelian subgroups of $G$ and obtain six centralizer types. Duflot's theorem gives depth at least three for four types, while exact ideal-quotient certificates exhibit regular sequences of length three for the remaining two. Hence every rank-two centralizer has cohomological depth at least three, so $H^*(G;k)$ has no associated prime of dimension two. The finite group presentation, the three cohomology-ring presentations, the enumeration summary, and the exact algebraic certificates are included for independent verification.

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.

This paper establishes a direct analogue of the classical Coding Theorem in the setting of symmetry groups. We consider computable bijections on the set of binary strings, called symmetries and define the symmetry prior of a string as the probability that a randomly chosen symmetry from a given group has the string as its unique fixed point. We show that for any fix-retractable symmetry group, a group admitting a computable section that selects an isolating symmetry for every string, the symmetry prior is a universal lower semi-computable semi-measure. In this case, the Geometric Coding Theorem holds. We also develop a Galois connection between subgroups of G and subsets of binary strings, characterizing closed points and maximal closed subgroups, and explore the join-semilattice of dense subgroups. Our results unify algorithmic information theory with group theory and provide a framework for studying symmetry-induced complexity measures. This paper is the first in a series on Computational Algorithmic Statistics (CAS).

Finite groups are rigid algebraic objects, whose Cayley graphs expose a rich network geometry through which group-theoretic structure can be measured, compared, and learned. In this paper, a dataset of $131{,}406$ Cayley graphs is constructed, covering all groups of order at most $767$ except order $512$, recording exact algebraic labels for group properties together with a broad collection of graph, cycle, distance, and spectral statistics. This census aims to provide novel benchmarks for studying how finite-group properties are reflected in Cayley graph observables. It also yields new enumerative contributions: alongside recovering known OEIS sequences for standard group classes, new sequences for monolithic groups and for groups generated by at most three, four, and five elements are contributed to the OEIS. The accompanying network analysis identifies several empirical regularities and formulates testable conjectures, including relationships involving square clustering, Cayley graph diameter, average graph disorder, and spectral eigengaps of nilpotent groups. Finally, a comparison between classical models, an MLP, and graph neural network architectures is performed for predicting algebraic group properties directly from Cayley graph data. The results show that engineered graph statistics are highly informative, while GNNs, especially GIN and in some fixed-order settings GCN, can recover substantial structural signal directly from the graph. Such that graph-aware architectures show phases of optimality on these group-theoretic graph representations.

Determining whether one finite group is isomorphic to a subgroup of another is a fundamental problem in computational group theory. In this work, we propose a Siamese Graph Neural Network (Siamese GNN) for subgroup prediction using Cayley graph representations of finite groups. Each input group is represented by its undirected Cayley graph and encoded by one branch of a Siamese GNN to produce a graph embedding. The resulting graph embeddings are combined with algebraic features derived directly from the input groups to construct a joint feature vector, which is processed by a fully connected classifier to predict subgroup relations between finite groups. By integrating graph-based structural representations with algebraic features, the proposed framework provides a unified approach for learning subgroup relations from finite groups. Experimental results demonstrate the effectiveness of the proposed architecture, achieving a test accuracy of 95.9% (47/49) on an independent test set and illustrating the potential of geometric deep learning for subgroup prediction.

A Graph Neural Network (GNN) framework for predicting the solvability of finite groups from their Cayley graph representations was introduced in [1]. In the present work, we generalize this approach and develop a property-independent framework for learning algebraic properties of finite groups directly from Cayley graphs. As representative case studies, we consider abelianity, nilpotency, and solvability. Using a common GNN architecture and training pipeline, we investigate the extent to which algebraic structure can be recovered from graph-based representations alone. Results on a collection of finite groups drawn from several families demonstrate that the framework successfully learns and distinguishes multiple algebraic properties from their associated Cayley graphs. These findings suggest that substantial algebraic information is encoded in graph representations and can be extracted through GNNs. More broadly, the proposed framework provides a proof of concept for applying graph representation learning to the study of algebraic properties of finite groups.

State tracking exposes a sharp limitation of sequence models: the relevant signal is often not a summary of observed tokens, but an ordered latent state that evolves through non-commutative transformations. We introduce a held-out transition-pair falsifier for finite non-Abelian group tracking. The protocol forbids selected ordered generator pairs during training and requires the same local patterns during evaluation, blocking one direct local-transition memorization pathway. In a controlled $S_3 \times S_3$ benchmark, a projected recurrent state model trained only on length-8 sequences produces error-free final-state predictions (perfect 250/250 per horizon) through evaluation horizons up to 1,048,576 tokens across five seeds. Matched native-readout baselines, including bag, GRU, and a single-configuration structured state-space model, remain near floor under the same protocol. Projection-matched GRU, structured SSM, and bag baselines equipped with analogous finite-group prototype readouts also remain near chance under the same split. Mechanism diagnostics show that hard projection coincides with low homomorphism error, low state-consistency drift, and non-trivial commutator separation, while softened projection collapses final-state accuracy. Clean-split audits verify zero verbatim reduced-word overlap and zero structural-template overlap between training and evaluation partitions. The evidence is scoped to this controlled finite-group falsifier rather than to a general architecture ranking. Within that regime, explicit projected non-commutative state composition acts as a useful inductive bias for long-horizon hidden-state tracking.

True general intelligence requires not only a model of the physical world but also a social world model: the capacity to infer how individual mental states interact and crystallize into group-level outcomes. Despite notable progress in individual-level Theory of Mind (ToM) reasoning, existing multimodal large language models fail at this broader task. Collective behavior emerges non-linearly from social tensions, conformity dynamics, and structural constraints, meaning it cannot be recovered by merely summing individual intentions. We present GroupToM-Bench, the first multimodal benchmark for group-level ToM, built around a causal chain spanning micro-level BDI states (belief, desire, intention), meso-level group tension and structural constraints, and macro-level outcome prediction and mechanistic attribution. To probe this full arc, we develop a seven-level cognitive audit framework. Experiments reveal a gap between current models and human baselines, highlighting a failure to process social structures and non-linear collective dynamics.

We introduce Lie groupoid equivariant neural networks as a specialization of recently proposed topological category-equivariant neural networks to the differentiable setting. Lie groupoid equivariant neural networks are composed from Lie groupoid lifting convolutions and Lie groupoid convolution layers, and we show how for suitable Lie groupoids they are equivalent to certain Lie algebroid-equivariant neural networks. We additionally describe groupoid invariant global pooling as a generalization of group invariant global pooling. Furthermore, we show that each of the aforementioned layers is a special case of recently introduced admissible category-equivariant layers by demonstrating that they define continuous natural transformations between continuous feature functors.

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

Free energy estimation is a fundamental yet challenging problem, from physics to statistics. Classical approaches rely on thermodynamic transformations, ranging from direct estimation, quasistatic integration, to finite-time averaging. Recent work [He and Du et al., 2025] learns neural transports to significantly accelerate the efficiency in the finite-time regime. In this paper, we generalize this framework to arbitrary state spaces. Building on this view, we develop a generalized neural transport learning approach for efficient estimation. Experiments validate the effectiveness and efficiency of the proposed method beyond continuous settings, extending to discrete and multimodal spaces as well as autoregressive settings. Beyond free energy estimation, we establish algebraic identities and reveal a group-theoretic structure linking infinitesimal time reversal and generalized Doob's $h$-transforms, showing that their compositions form a generalized dihedral group.

A finite presentation < X | R > of a finite group is called `just finite' if removing any relation from R results in a presentation for an infinite group. It has been an open question (Kourovka Notebook, Problem 21.10) whether every finite group admits such a presentation. We resolve this conjecture in the affirmative.

We prove lower bounds on learning the Möbius or Liouville function with a variety of standard learning techniques, including kernel methods, noisy gradient methods, and correlational statistical query algorithms. These results follow from quantitative bounds on the correlation of Möbius with digital characters of various finite abelian groups, where the group is dictated by the type of input data the algorithm is given. Using residues mod $p$ for many different primes corresponds to a cyclic group, and using the base $p$ expansion for a fixed prime corresponds to an elementary abelian $p$-group. We also note that lower bounds of this form are closely related to certain types of digital prime number theorems.

While current deep learning models achieve high performance by learning statistical correlations from vast datasets,which stands in stark contrast to human learning. They lack the flexibility of humans-particularly preverbal infants-to autonomously acquire the underlying structure of the world from limited experience and adapt to novel situations. In this study, we propose an unsupervised representation learning method based on a hierarchical relationship in group operations, rather than statistical independence, aiming to build a computational model of the cognitive development of infants. The proposed model features an integrated architecture that simultaneously performs object segmentation and the extraction of motion laws from dynamic image sequences. By introducing the Homomorphism from algebra as a structural constraint within a neural network, the model structurally separates pixel-level changes into meaningful, decomposed transformation components, such as translation and deformation. Using interaction scenes (chasing and evading tasks) based on developmental science findings, we experimentally demonstrate that the model can segment multiple objects into individual slots without any ground-truth labels. Furthermore, we confirmed that relative movements between objects, such as approaching or receding, are accurately mapped and structured into a one-dimensional additive latent space. These results suggest that by introducing algebraic geometric constraints rather than relying solely on statistical correlation learning, physically interpretable "disentangled representations" can be acquired. This study contributes to the understanding of the process by which infants internalize environmental laws as structures and provides a new perspective for constructing artificial systems with developmental intelligence.

The higher-order autocorrelations of integer-valued or rational-valued functions on finite Abelian groups appear naturally in X-ray crystallography, and have applications in computer vision systems, correlation tomography, correlation spectroscopy, and pattern recognition. In this paper, we consider the problem of reconstructing a rational-valued function on finite Abelian groups from its higher-order autocorrelations. We describe an explicit reconstruction algorithm, and prove that the autocorrelations up to order $3r+3$ are always sufficient to determine the data up to translation, where $r$ is the rank of the group. We also provide examples of rational-valued functions on finite Abelian group which are not determined by their autocorrelations up to order $3r+2$. In particular, we provide a sharp upper bound on the separating degree of the regular representation of a finite Abelian group in terms of its rank.