研究有限群与域扩张中极大Chowla集，揭示其结构与阶数关系。
Extremal Chowla sets and their linear analogues: A human-AI mathematical investigation using Co-Scientist
- 定义并分析有限群中Chowla集的最大大小，关联元素阶分布。
- 对循环群给出精确公式，证明其与φ(n)比值的上下极限分别为1和无穷。
- 推广至域扩张，建立线性Chowla子空间的精确表达式，适合代数数论研究者。
我们引入一个与有限群中Chowla型阶条件相关的极值不变量。若有限群$G$的非空子集$S$中每个元素的阶均大于$|S|$，则称$S$为Chowla集，并记$C(G)$为此类集合的最大基数。首先证明$C(G)$由$G$中元素阶的分布决定。对循环群，推导出基于因数的精确公式，并刻画了满足$C(\mathbb{Z}/n\mathbb{Z})=φ(n)$的整数$n$。证明$\liminf_{n\to\infty}C(\mathbb{Z}/n\mathbb{Z})/φ(n)=1$，而$\limsup_{n\to\infty}C(\mathbb{Z}/n\mathbb{Z})/φ(n)=\infty$，并确定以$n$归一化下的上下极限。对有限交换群，基于不变因子分解得到显式公式，且给出有限交换$ p $-群的闭式解。随后发展了有限域扩张的线性类比：若非零$K$-子空间$A \subseteq L/K$满足对所有非零$a\in A$有$[K(a):K] > \dim_K A$，则称$A$为Chowla子空间。尽管该条件不强制每个元素生成全扩张，但在$L/K$有限可分时，证明$C(L/K)=[L:K]-d_{\max}(L/K)$，其中$d_{\max}(L/K)$是最大真中间域的度数。对有限域，通过正规基构造给出了任意次数的直接证明。本工作通过专家引导的人机协作完成，使用了聚焦推理的Co-Scientist配置探索例子与证明策略。作者独立验证并完成全部论证，撰写最终证明。
原文摘要 · Abstract (English)
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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