Why the non-abelian Gyarmati--Matolcsi--Ruzsa inequality does not generally work: mechanisms and explicit examples

For finite sets in a commutative semigroup, Gyarmati, Matolcsi and Ruzsa proved \[ |A_1++A_n|ⁿ⁻¹\ ≤\ ∏ᵢ₌₁^n|A_1++Aᵢ₋₁+Aᵢ₊₁++A_n| . \] They asked whether in an arbitrary group \[ |A_1⋯ A_n|ⁿ⁻¹\ ≤\ ∏ᵢ₌₁^n\ maxa∈ A_i\,|A_1⋯ Aᵢ₋₁\,a\,Aᵢ₊₁⋯ A_n| , \] where each omitted set is replaced by the largest slice obtained by fixing one of its elements. Nair, Namayandeh and Zhu proved that this inequality does not hold when $n=4$ and G=D₄¹⁴, of group order 8¹⁴. We develop its double-coset mechanism and construct counterexamples in many other groups. For $n=4$, the smallest group found has order $518\,400$, namely A₆²× C₂²; no absolute minimality is claimed. For $n=6$ there is a witness in a group of order $1728$, for $n=10$ a class-three $3$-group of order $243$ suffices, and for $n=4$ a single class-two $2$-group of order 2²³ works. We also obtain torsion-free examples. The violation ratio Rₙ=|A₁⋯ Aₙ|ⁿ⁻¹/∏ᵢNᵢ is greater than one exactly when the inequality fails; explicit four-set constructions attain R₄≈1.29 in A₈², and a larger ratio of about $1.43$ is reported for A₁₀². For every n≥4, the inequality holds in all direct powers of G precisely when G is abelian. For n≥2 the optimal exponent θₙ is the infimum of log(∏ᵢNᵢ)/log|A₁⋯ Aₙ| over products of size at least two. We obtain 2≤θₙ≤ n-1.0768, and an infinite family in C₃*C₃ shows θₙ≤ n-1-δ for every δ<5-2log₂5≈0.3561 and all sufficiently large n.