In this paper, 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 denote the maximum cardinality of such a set by $\\Ccal(G)$. We first show that $\\Ccal(G)$ 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 $\\Ccal(\mathbb{Z}/n\mathbb{Z})=\varphi(n)$. We prove that $\\liminf_{n\to\infty}\Ccal(\mathbb{Z}/n\mathbb{Z})/\varphi(n)=1$, while $\\limsup_{n\to\infty}\Ccal(\mathbb{Z}/n\mathbb{Z})/\varphi(n)=\infty$, determining 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, along 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 $\\Ccal(L/K)=[L:K]-d_{\max}(L/K)$, where $d_{\max}(L/K)$ is the largest degree over $K$ of a proper intermediate field. For finite fields, we provide a direct proof in every degree using a normal-basis construction. This work was developed through an expert-guided human--AI collaboration, utilizing a reasoning-focused configuration of Co-Scientist to explore examples and potential proof strategies. The authors independently verified and completed all arguments and wrote the final proofs.
Blogger's Review: This paper showcases profound mathematical structures through the introduction of extremal Chowla sets and their linear analogues in finite groups and fields. The human-AI collaborative approach opens new avenues for mathematical research, emphasizing the potential of AI in exploring new theories and proof strategies. Its results not only enrich the theoretical foundation of group theory but also offer significant insights for future research.