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Statement

Proposition 4.7 (p. 15). Let $r_2,\dots,r_5$ be pairwise coprime integers with $r_2$ odd. Then for any group $G$ the $5$-tuple $(3,3r_2,6r_3,6r_4,6r_5)$ is not $G$-harmonic: there are no subgroups $U_1,\dots,U_5$ with $[G:U_1]=3$, $[G:U_2]=3r_2$, $[G:U_j]=6r_j$ for $3\le j\le5$, and elements $g_i$ with $g_1U_1,\dots,g_5U_5$ 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 states (p. 15) that it knows no group realizing the configuration of Lemma 4.6, the general $5$-tuple $(3r_1,3r_2,6r_3,6r_4,6r_5)$ with $r_1,\dots,r_5$ pairwise coprime and $r_1,r_2$ odd, and that such a group would answer Question 1 negatively; Proposition 4.7 settles only the case $r_1=1$.

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 and its proof are on p. 15.

Read depth. Claims checked: the statement was read clause by clause against the print. The proof and Lemma 4.6 (pp. 11--14) were read for their structure only; nothing here is independently reviewed.

Proof pointer

p. 15, using Lemma 4.6 (p. 11, proved on pp. 11--14), which fixes, up to symmetry, every $\alpha(x,y)$ of a harmonic $5$-tuple of this shape and gives $3\mid r_2$ and $2\mid r_3$. The action of $G$ on the three cosets of $U_1$ gives a homomorphism to $S_3$; comparing the images of $U_1,U_2,U_4$ and $U_3$ with the product-set sizes from Lemma 4.6 shows each image has order $2$, and then two computations of $[G:U_2\cap U_3]$ force $[U_3\cap N:U_3\cap U_2]=r_2/2$, with $N$ the kernel, which is not an integer since $r_2$ is odd.

Dependencies

Lemma 4.6 of the same paper, and through it Lemmas 3.1, 3.4, 3.5, 3.7 and 3.10, Corollaries 3.6 and 3.11, Propositions 3.8 and 4.2.

Bears on

• Problem 274: an input to the proof of Theorem A, needed only for groups of order $1080$, to exclude index sets containing $\{3,6,9,12,30\}$. It is one of the four obstructions the Itabe claim page lists under Depends on.