A Finite E-Group of Nilpotency Class Three

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 $\mathrm{PG}(8,3)$.

An Exact Counterexample to Carlson's Associated-Prime Depth Conjecture from a Group of Order 128

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.

Extremal Chowla sets and their linear analogues: A human-AI mathematical investigation using Co-Scientist

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.

Learning Subgroup Relations Using Siamese Graph Neural Networks

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 General Framework for Learning Algebraic Properties from Cayley Graphs using Graph Neural Networks

In this work, we present a general Graph Neural Network (GNN) framework for learning algebraic properties of finite groups from their Cayley graph representations. The framework provides a unified computational pipeline consisting of a common graph construction procedure, feature representation, training methodology, and GNN architecture, with only the target labeling function varying across classification tasks. To demonstrate the generality of the proposed approach, we consider three representative algebraic properties: abelianity, nilpotency, and solvability. Experiments were conducted on a benchmark of 176 finite groups drawn from several classical families, with all groups included in each classification task. To address class imbalance, class-weighted cross-entropy loss was employed where appropriate during training. Furthermore, the family PSL(2,q) was reserved exclusively for testing, enabling evaluation of the framework's ability to generalize to previously unseen group families. The best-performing models achieved test balanced accuracies of 1.000, 0.856, and 0.875 for abelianity, nilpotency, and solvability, respectively. Although the same computational framework was employed across all tasks, different GNN architectures proved optimal for different algebraic properties, suggesting that the representational complexity required to learn a property depends on its underlying algebraic structure. These results demonstrate that GNNs can effectively learn multiple algebraic properties directly from Cayley graph representations while exhibiting strong generalization to unseen group families. More broadly, the proposed framework establishes a computational methodology for studying algebraic properties of finite groups using graph-based machine learning.