发表机构
Clayton State University; Google Cloud AI Research; Google DeepMind（克莱顿州立大学; 谷歌云人工智能研究部; 谷歌DeepMind）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究有限群和有限域扩张中与乔拉型序条件相关的极值不变量，通过专家指导的人机合作，利用联合科学家探索示例与策略，得出循环群、有限阿贝尔群及有限域扩张相关公式及结论。
AI 中文摘要
我们引入了与有限群中乔拉型序条件相关的一个极值不变量。有限群\(G\)的非空子集\(S\)若满足\(S\)中每个元素的阶大于\(|S|\)，则称\(S\)为乔拉集，记\(\Ccal(G)\)为此类集合的最大基数。首先表明\(\Ccal(G)\)由\(G\)中元素阶的分布决定。对于循环群，得出精确的除数公式并刻画满足\(\Ccal(\mathbb Z/n\mathbb Z)=\varphi(n)\)的整数\(n\)。证明了\(\liminf_{n\to\infty}\Ccal(\mathbb Z/n\mathbb Z)/\varphi(n)=1\)，\(\limsup_{n\to\infty}\Ccal(\mathbb Z/n\mathbb Z)/\varphi(n)=\infty\)，并确定了归一化后的相应上下限。对于有限阿贝尔群，根据不变因子分解得到显式公式，以及有限阿贝尔\(p -\)群的封闭公式。接着为有限域扩张发展了线性类似物。有限域扩张\(L/K\)的非零\(K -\)子空间\(A\)，若对每个非零\(a\in A\)有\([K(a):K]>\dim_K A\)，则称\(A\)为乔拉子空间。当\(L/K\)有限可分时，证明了精确公式\(\Ccal(L/K)=[L:K]-d_{\max}(L/K)\)。对于有限域，利用正规基构造在每个次数下给出直接证明。这项工作是通过专家指导的人机合作开展的。使用聚焦推理的联合科学家配置来探索示例和潜在证明策略。作者提出问题，独立验证并完成所有论证，写出最终证明。
英文摘要
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 $C(\mathbb{Z}/n\mathbb{Z})=φ(n)$. We prove that $\liminf_{n\to\infty}C(\mathbb{Z}/n\mathbb{Z})/φ(n)=1$, whereas $\limsup_{n\to\infty}C(\mathbb{Z}/n\mathbb{Z})/φ(n)=\infty$, 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.
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