Two Questions on $G$-harmonic Tuples

Abstract

An $n$-tuple of positive integers is $G$-harmonic if there are subgroups of $G$ having those indices whose cosets can be chosen pairwise disjoint, and $\mathbb{Z}$-harmonic if there are pairwise disjoint residue classes with those moduli. Ginosar asked whether every $G$-harmonic tuple is $\mathbb{Z}$-harmonic. Margolis and Schnabel proved this for tuples of length at most $4$, and analysed a particular family of length-$5$ tuples that would yield a counterexample if any member were $G$-harmonic. We show that the bound $4$ is sharp: $(6,6,6,10,15)$ is $A_5$-harmonic but not $\mathbb{Z}$-harmonic. Moreover, the five pairwise disjoint cosets realising this tuple can be extended to a coset partition of $A_5$ using only cosets of indices $6$, $10$, and $15$. The index tuple of this partition is not $\mathbb{Z}$-harmonic; because its indices repeat, this does not contradict the Herzog--Schönheim conjecture. We also prove that no member of the length-$5$ family analysed by Margolis and Schnabel in connection with possible counterexamples is $G$-harmonic for any group $G$.

Disclosure

“r’s ques- tion negatively, was obtained by Anthropic’s Claude Opus 5 under the author’s prompt- ing. The proof that the Lemma 1.1 configuration is realised in no group (Theorem 4.2) was obtained by Z.ai’s GLM-5.2, likewise under prompting. Claude Opus 5 and GLM-5.2 were also prompted to critique and cross-check each other’s arguments; no transcript of this exchange was retained, but the corrections it produced are reflected in the final proofs. Computations were performed and independen”

PDF page 7

Classification

Substantial proof generation

Multiplier

10

Verified

2026-08-23T05:18:25.663606Z

Structural counts