---

Claim. The Herzog-Schönheim conjecture holds for every group $G$ of order less than $1440$: whenever such a group is partitioned into two or more cosets $g_iU_i$, two of the cosets come from subgroups of the same index. This is [[../library/covering_systems/margolis_2019_herzog_schonheim_conjecture_small_groups/theorem_a|Theorem A]] of L. Margolis and O. Schnabel, The Herzog-Schönheim conjecture for small groups and harmonic subgroups, Beitr. Algebra Geom. 60 (2019), no. 3, 399--418, extending Ginosar's earlier bound of $240$. The proof restricts the indices of a minimal counterexample (pairwise distinct, reciprocal sum $1$, any two with a common divisor, none equal to $2$), proves in Theorem B that every $G$-harmonic tuple of length at most four is $\mathbb Z$-harmonic, and excludes the remaining index tuples by Propositions 4.2, 4.3, 4.5 and 4.7. The statements are recorded on the library's source card.

Covers. The case of Problem 274 for groups of order below $1440$: no such group has an exact covering by two or more cosets of pairwise different sizes. The question for larger finite groups, and so for infinite groups, stays open.

Depends on. Nothing in this wiki; the theorem is the paper's own.

Acceptance. Refereed: Beiträge zur Algebra und Geometrie 60 (2019), no. 3, 399--418, doi:10.1007/s13366-018-0419-1, published online 2018-10-04; the arXiv v1 of 2018-03-09 names this page. Not reviewed: the site's commentary credits the theorem, but the site labels the problem OPEN, so the credit is not counted as review. Not formalized: no Lean proof of the theorem is recorded.