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Statement

Proposition 4.3 (p. 8). Let $r_1,r_2,r_3,r_4$ be pairwise coprime integers. Then for any group $G$ the $4$-tuple $(3r_1,3r_2,3r_3,3r_4)$ is not $G$-harmonic: there are no subgroups $U_i$ with $[G:U_i]=3r_i$ and elements $g_i$ with $g_1U_1,\dots,g_4U_4$ pairwise disjoint.

The $r_i$ are positive, since a $G$-harmonic tuple consists of indices. The statement falls under the paper's convention, from Section 3 on (p. 5), that every group is finite; the reduction recorded on the Theorem B page extends it to groups in general. The paper notes (p. 8) that Zhu proved it earlier ([Zhu08, Theorem 3.1]).

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 proposition is on p. 8, its proof on pp. 8--9.

Read depth. Claims checked: the statement was read clause by clause against the print. The proof was read for its structure only; nothing here is independently reviewed.

Proof pointer

pp. 8--9. Write $[G:U_i\cap U_j]=\operatorname{lcm}([G:U_i],[G:U_j])\,\alpha(i,j)$ (Notation 3.3, p. 5). Disjointness forces $\alpha(i,j)\le2$. Two claims, using Corollary 3.11 and Lemma 3.10, show that for any three of the subgroups, the number of their three pairs with $\alpha=2$ is zero or three; Proposition 3.8 then rules out the all-ones pattern, so every $\alpha(i,j)=2$. In that case the product sets $U_1U_2,U_1U_3,U_1U_4$ each have size $2|G|/3$ and pairwise intersections of size $|G|/3$, and the counting identity of Lemma 3.1 makes their common intersection empty, though it contains $U_1$.

Dependencies

Lemmas 3.1, 3.4 and 3.10, Corollaries 3.6 and 3.11, and Proposition 3.8 of the same paper.

Bears on

• Problem 274: an input to the proofs of Theorem B (one of the two $4$-tuple forms) and Theorem A, where it excludes index sets containing $\{3,6,9,15\}$. It is one of the four obstructions the Itabe claim page lists under Depends on.