Two Questions on $G$-harmonic Tuples

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Murali Menon, "Two Questions on $G$-harmonic Tuples," arXiv:2608.15873 (2026).

The copy read for this card is the held arXiv v1 PDF; a complete Markdown reading copy sits beside it. The arXiv v1 source carries the paper's ancillary GAP script as anc/verify-counterexample.g. The arXiv record (https://arxiv.org/abs/2608.15873, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

An $n$-tuple $(a_1,\ldots,a_n)$ is called $G$-harmonic when there are subgroups $U_i\leq G$ of indices $a_i$ and pairwise disjoint cosets $g_iU_i$. It is $\mathbb Z$-harmonic when one can choose residues $r_i\pmod {a_i}$ whose classes are pairwise disjoint, equivalently

$$r_i\not\equiv r_j\pmod {\gcd(a_i,a_j)}\qquad(i\ne j).$$

The paper answers negatively the question whether every $G$-harmonic tuple is $\mathbb Z$-harmonic, and it separately proves that the special five-subgroup configuration proposed by Margolis and Schnabel cannot produce such a counterexample.

Located results

Lemma 2.1 (Section 2, "Preliminaries"). Let $(a_1,\ldots,a_n)$ be a tuple of positive integers.

• (i) If three entries equal $6$, some entry $x$ has $\gcd(6,x)=2$, and some entry $y$ has $\gcd(6,y)=3$, then the tuple is not $\mathbb Z$-harmonic.
• (ii) If four entries equal $6$ and some entry $x$ has $\gcd(6,x)=2$, then the tuple is not $\mathbb Z$-harmonic.

For the first clause, the three residues belonging to the $6$'s must be the three residues of one parity modulo $6$, and hence exhaust all residue classes modulo $3$; the $y$-residue then has no available class. For the second, one parity contains only three residue classes modulo $6$, so it cannot contain four distinct $6$-residues.

Theorem 3.1 (Section 3, "Answer to Question 1"). The tuple $(6,6,6,10,15)$ is $A_5$-harmonic but not $\mathbb Z$-harmonic. The latter claim is exactly Lemma 2.1(i), since $\gcd(6,10)=2$ and $\gcd(6,15)=3$. The $A_5$ witness is the following explicit family. Put

$$\begin{aligned} D&=\langle(1\,2\,5\,3\,4),(1\,2)(4\,5)\rangle, &|D|&=10,&[A_5:D]&=6,\\ T&=\langle(2\,3\,5),(1\,4)(2\,3)\rangle, &|T|&=6,&[A_5:T]&=10,\\ V&=\langle(1\,2)(3\,4),(1\,3)(2\,4)\rangle, &|V|&=4,&[A_5:V]&=15. \end{aligned}$$

Then the five cosets

$$(1\,5\,3)D,\quad (1\,4\,2\,3\,5)D,\quad (1\,5\,2\,3\,4)D,\quad (1\,2\,4)T,\quad (3\,4\,5)V$$

are pairwise disjoint. Their sizes are $10,10,10,6,4$, so their union has $40$ of the $60$ elements of $A_5$ and their index tuple is $(6,6,6,10,15)$.

Corollary 3.2 (Section 3.1, "Extension to a coset partition"). Adjoin

$$(1\,2)(3\,4)V,\quad (1\,2)(4\,5)V,\quad (1\,4\,5)T_2,\quad (1\,4\,3\,2\,5)T_3,$$

where

$$T_2=\langle(1\,3\,4),(1\,3)(2\,5)\rangle, \qquad T_3=\langle(1\,2\,4),(1\,2)(3\,5)\rangle.$$

These four cosets and the five from Theorem 3.1 are pairwise disjoint and partition $A_5$. The index multiset is

$$\{6,6,6,10,10,10,15,15,15\},$$

or the repeated-index profile $(6^3,10^3,15^3)$, with $3/6+3/10+3/15=1$. Its tuple is again not $\mathbb Z$-harmonic by Lemma 2.1(i).

Theorem 4.2 (Section 4, "The Lemma 1.1 configuration is impossible"). If $r_1,\ldots,r_5$ are pairwise coprime positive integers and $r_1,r_2$ are odd, then for every group $G$ the tuple

$$(3r_1,3r_2,6r_3,6r_4,6r_5)$$

is not $G$-harmonic. Margolis and Schnabel's Lemma 4.6 would force, after relabelling, $\alpha(U_3,U_4)=3$, $\alpha(U_3,U_5)=\alpha(U_4,U_5)=1$, and $3\mid r_2$. Pairwise coprimality then gives $3\nmid r_3r_4r_5$, while Proposition 4.1 rules out precisely those three $\alpha$-values. Thus the pattern that had been proposed as a possible counterexample cannot occur in any group.

GAP appendix locator. Appendix A, "GAP verification" (p. 8), defines the five subgroups D, T, V, T2, and T3 and the helper Coset. The block headed "the five pairwise-disjoint cosets of Theorem 3.1" constructs L5, returns subgroup indices [ 6, 10, 15 ], ten zero pairwise-intersection sizes, and union size 40. The immediately following block headed "extension to a coset partition, profile (6^3,10^3,15^3) (Corollary 3.2)" constructs L9, returns the sorted index list [6,6,6,10,10,10,15,15,15], checks every pairwise intersection is empty, and checks that the union equals Set(Elements(G)). The appendix warns that GAP composes permutations left-to-right: its set $\{ug:u\in H\}$ represents the displayed left coset $gH$.

Relation to Problem 274

Corollary 3.2 is the closest near-counterexample here to Problem 274: unlike the partial family in Theorem 3.1, it is a genuine exact coset partition, and unlike an integer covering-system model, its index tuple is not $\mathbb Z$-harmonic. It still does not answer Problem 274. Each index occurs three times, so the partition has only three coset sizes, each repeated; Problem 274 asks for a nontrivial exact covering whose cosets have different sizes, equivalently the distinct-index case addressed by the Herzog--Schönheim conjecture. Theorem 3.1 itself covers only $40/60=2/3$ of $A_5$, and Theorem 4.2 excludes only its stated pairwise-coprime $(3r_1,3r_2,6r_3,6r_4,6r_5)$ pattern. None of these facts rules out or constructs an arbitrary distinct-index partition.

Ideas that transfer to work on Problem 274 are:

• pass any finite family of finite-index subgroups to the finite quotient by the intersection of their cores; indices, subgroup intersections, and coset disjointness are preserved;
• search first for pairwise-disjoint cosets with the desired indices and then impose the exact-cover equation $\sum_i1/[G:U_i]=1$;
• combine the product formula $|UV|=|G|[G:U\cap V]/([G:U][G:V])$ with integrality and containment of products such as $U_3(U_4\cap U_5)$ to exclude proposed intersection profiles; and
• retain explicit subgroup generators and representatives so a candidate can be replayed independently in GAP or another finite-group system.

The non-$\mathbb Z$-harmonic residue obstruction does not itself obstruct a group coset partition; Corollary 3.2 is exactly a counterexample to that transfer. Nor may repeated-index searches or the five special tuples in Section 5 be treated as evidence for the distinct-index case.

Reading and verification status

Read status: claims checked. The definitions and exact statements of Lemma 2.1, Theorem 3.1, Corollary 3.2, and Theorem 4.2 were checked clause by clause against the held PDF. Their proofs have not been independently verified here.

The source explicitly reports that Anthropic's Claude Opus 5, under Menon's prompting, obtained the $A_5$ realization in Theorem 3.1, and that Z.ai's GLM-5.2, likewise prompted, obtained the impossibility proof in Theorem 4.2. The two models were prompted to critique and cross-check one another; no transcript was retained, and the resulting corrections were incorporated into the paper. The author reports that the computations were done, and independently cross-checked, in GAP and in custom code, and accepts responsibility for the mathematics.

That is source-reported provenance, not an independent repository replay. The ancillary script ships with the arXiv v1 source, but no run of it is recorded here. Before consuming the computational assertions as independently checked results, a reviewer should run that script, or the Appendix A excerpt, in GAP and independently reproduce the L5 and L9 intersection and coverage checks. Theorem 4.2's argument likewise still needs independent mathematical review.

Bears on. The exact-partition construction is qualified near-counterexample context for Problem 274, not a resolution of it.