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(Z/nZ)=φ(n). We prove that
liminfn→∞C(Z/nZ)/φ(n)=1, whereas
limsupn→∞C(Z/nZ)/φ(n)=∞, 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]>dimKA for every nonzero
a∈A. Since this condition depends on
dimKA, 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]−dmax(L/K), where
dmax(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.