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Statement

Proposition 4.5 (p. 10). Let $r_1,r_2,r_3,r_4$ be pairwise coprime integers with $r_1$ odd. Then for any group $G$ the $4$-tuple $(2r_1,4r_2,4r_3,4r_4)$ is not $G$-harmonic: there are no subgroups $U_i$ with $[G:U_1]=2r_1$, $[G:U_i]=4r_i$ for $2\le i\le4$, 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.

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 and its proof are on p. 10.

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

p. 10, using Lemma 4.4 (p. 9). With $\alpha(i,j)$ as in Notation 3.3 (p. 5), Lemma 4.4 gives $\alpha(1,i)=1$ and $\alpha(i,j)\in\{1,3\}$ for $2\le i,j\le4$. Proposition 3.8 and Corollary 3.6 force $\alpha(2,3)=\alpha(2,4)=\alpha(3,4)=3$, and then $3\mid r_1$. The product sets $U_2U_1,U_2U_3,U_2U_4$ then cover $G$ in pairs, and Lemma 3.1 makes their common intersection empty, though it contains $U_2$.

Dependencies

Lemmas 3.1, 3.4, 4.1 and 4.4, Corollary 3.6 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 is needed only for groups of order $720$, to exclude index sets containing $\{4,6,8,20\}$. It is one of the four obstructions the Itabe claim page lists under Depends on.