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Statement

Proposition 4.2 (p. 8). Let $r_1,r_2,r_3$ be pairwise coprime integers. Then for any group $G$ the $3$-tuple $(2r_1,2r_2,2r_3)$ is not $G$-harmonic: there are no subgroups $U_1,U_2,U_3$ with $[G:U_i]=2r_i$ and elements $g_i$ with $g_1U_1,g_2U_2,g_3U_3$ 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 2.2]).

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.

Read depth. Claims checked: the statement was read clause by clause against the print and the short proof was read; nothing here is independently reviewed.

Proof pointer

p. 8. Lemma 4.1 (p. 8) shows that two subgroups of indices $2r_1,2r_2$ with $r_1,r_2$ coprime and disjoint cosets satisfy $[G:U_1\cap U_2]=2r_1r_2$. Applied to each pair, this meets the hypotheses of Corollary 3.9 (p. 7), which says such a tuple is not harmonic.

Dependencies

Lemma 4.1, Corollary 3.9, Proposition 3.8 and Lemma 3.5 of the same paper; Lemma 2.2 (after [GS11, Corollary 2.1]).

Bears on

• Problem 274: an input to the proofs of Theorem B ($n=3$) and Theorem A, where it excludes index sets containing $\{4,6,10\}$ or $\{4,6,14\}$. It is one of the four obstructions the Itabe claim page lists under Depends on.