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Statement

Definitions (p. 2). An $n$-tuple $(a_1,\dots,a_n)$ of positive integers is $\mathbb Z$-harmonic if there are integers $m_1,\dots,m_n$ such that the progressions $m_1+a_1\mathbb Z,\dots,m_n+a_n\mathbb Z$ are pairwise disjoint (the paper says "have pairwise trivial intersection"). For a group $G$, it is $G$-harmonic (Definition 1.1) if there are subgroups $U_1,\dots,U_n$ of $G$ with $[G:U_i]=a_i$ for every $i$ and elements $g_1,\dots,g_n\in G$ such that the cosets $g_1U_1,\dots,g_nU_n$ are pairwise disjoint.

Theorem B (p. 2). "Let $G$ be a group and let $(a_1,\dots,a_n)$ be a $G$-harmonic tuple, where $n\le4$. Then $(a_1,\dots,a_n)$ is also $\mathbb Z$-harmonic." Equivalently, Ginosar's Question 1 (p. 2), whether every $G$-harmonic $n$-tuple is $\mathbb Z$-harmonic, has a positive answer for every group $G$ when $n\le4$.

Scope of the proof. From Section 3 on, the paper's convention is that every group is finite (p. 5), and the proof of Theorem B (pp. 10--11) works under it. The infinite case follows by a reduction recorded here, not in the paper: the subgroups in a $G$-harmonic tuple have finite index, so the intersection $N$ of their normal cores has finite index; each coset $g_iU_i$ is a union of cosets of $N$, so the images in the finite group $G/N$ are pairwise disjoint cosets of subgroups with the same indices, and the tuple is $G/N$-harmonic.

The paper calls the theorem a generalization of Zhu's confirmation, for $n\le4$, of Sun's question whether a $G$-harmonic $n$-tuple must contain two entries with greatest common divisor at least $n$ (p. 2, citing [Zhu08] and [Sun06, Conjecture 1.2]). For $n=2$ it is the second sentence of the paper's Lemma 2.2 (p. 3, after [GS11, Corollary 2.1]): two coprime integers never form a $G$-harmonic pair.

Source. L. Margolis and O. Schnabel, The Herzog-Schönheim conjecture for small groups and harmonic subgroups, Beitr. Algebra Geom. 60 (2019), no. 3, 399--418, doi:10.1007/s13366-018-0419-1. Labels and pages are those of arXiv:1803.03569v1, the edition the source card names: the statement is on p. 2, the proof on pp. 10--11.

Read depth. Claims checked: the statement and both definitions were read clause by clause against the print. The proof was read for its structure only, and the cited results of Sun on $\mathbb Z$-harmonic tuples were not checked; nothing here is independently reviewed.

Proof pointer

Proof of Theorem B, pp. 10--11. It suffices to treat a tuple that is not $\mathbb Z$-harmonic while all its proper sub-tuples are. For $n=2$ the two entries are coprime and Lemma 2.2 applies. For $n=3$, Sun's answer to Problem 2 of Huhn and Megyesi forces the form $(2r_1,2r_2,2r_3)$ with the $r_i$ pairwise coprime, excluded by Proposition 4.2. For $n=4$, Sun's answer to their Problem 1 leaves the forms $(2r_1,4r_2,4r_3,4r_4)$ with $r_1$ odd and $(3r_1,3r_2,3r_3,3r_4)$, with the $r_i$ pairwise coprime, excluded by Proposition 4.5 and Proposition 4.3.

Dependencies

Lemma 2.2 and Propositions 4.2, 4.3 and 4.5 of the same paper; Z.-W. Sun, Solutions to two problems of Huhn and Megyesi, Chinese Ann. Math. Ser. A 13 (1992), 722--727 (the paper's [Sun92]), which settles Problems 1 and 2 of Huhn and Megyesi, Discrete Math. 41 (1982) ([HM82]), for $4$- and $3$-tuples.

Bears on

• Problem 274: the paper notes (p. 2) that a positive answer to Question 1 for all $n$ would, with the Davenport-Mirsky-Newman-Rado theorem, give the Herzog-Schönheim conjecture. Theorem B gives that answer only for $n\le4$; by the same reasoning, recorded here rather than in the paper, it excludes exact coverings by two to four cosets of pairwise different indices. The paper calls it a key ingredient in the proof of Theorem A (p. 2), whose proof cites Propositions 4.2, 4.3 and 4.5 directly. It does not decide the problem.