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Statement

Theorem A (p. 1). "Any group of order less than 1440 satisfies the Herzog-Schönheim Conjecture."

The conjecture, as the paper poses it on p. 1 after Herzog and Schönheim (1974): let $\{g_iU_i\}_{i=1}^n$ be a non-trivial partition of a group $G$ into cosets, where $U_1,\dots,U_n$ are subgroups of finite index in $G$; then the indices $[G:U_1],\dots,[G:U_n]$ are not pairwise distinct. The abstract (p. 1) restates the theorem as the existence of distinct $1\le i,j\le n$ with $[G:U_i]=[G:U_j]$ in every non-trivial coset partition of $G$, so a non-trivial partition is one with $n\ge2$ cosets.

So for every group $G$ with $|G|<1440$ and every partition $G=g_1U_1\sqcup\dots\sqcup g_nU_n$ into $n\ge2$ left cosets of subgroups, two of the subgroups have the same index. Ginosar had proved this for $|G|<240$ (the paper's [Gin18]); the paper notes (p. 3) that for $|G|=240$ the arithmetic conditions of Lemma 2.3 alone no longer suffice.

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 statement is on p. 1, the proof on pp. 15--16.

Read depth. Claims checked: the statement and the conjecture it refers to were read clause by clause against the print. The proof was read for its structure only; its computer enumeration of Egyptian fractions was not repeated, and nothing here is independently reviewed.

Proof pointer

Proof of Theorem A, pp. 15--16. For a partition without repeated indices of a counterexample of order below $1440$, taken of least order, Lemma 2.3 and Lemma 2.6 (p. 4) make the indices $a_i$ pairwise distinct, greater than $2$, with $\sum1/a_i=1$, any two sharing a factor, and their common divisor divisible by $2$ or by $3$. What the paper calls "elementary computer calculations" list the group orders below $1440$ admitting such index sets made of divisors of $|G|$; in each case the proof finds among the indices a sub-tuple excluded by one of Proposition 4.2 (for instance $\{4,6,10\}$ or $\{4,6,14\}$), Proposition 4.3 ($\{3,6,9,15\}$), Proposition 4.5 ($\{4,6,8,20\}$, needed only for $|G|=720$) or Proposition 4.7 ($\{3,6,9,12,30\}$, needed only for $|G|=1080$). The cosets of a partition are pairwise disjoint, so a sub-tuple of the indices is $G$-harmonic, which those propositions forbid.

Dependencies

Lemma 2.3, Lemmas 2.4--2.6 (pp. 3--4) and Propositions 4.2, 4.3, 4.5 and 4.7 of the same paper; Ginosar and Schnabel (J. Comb. Number Theory 3 (2011), the paper's [GS11]) for Lemma 2.3 d) and the reduction in Lemma 2.4; Burnside's $p$-complement theorem and the conjecture for pyramidal groups (Berger, Felzenbaum and Fraenkel, the paper's [BFF87]) in Lemma 2.5.

Bears on

• Problem 274: for a finite group, cosets of different sizes are cosets of subgroups of different indices, so the theorem says that no group of order below $1440$ has an exact covering by two or more cosets of pairwise different sizes. It says nothing about larger groups. The problem page records the theorem on the Margolis and Schnabel claim page.