Margolis 2019 herzog schonheim conjecture small groups

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lemma_2_3: The indices of a coset partition with no repeated index are pairwise distinct, have reciprocals summing to 1 and share a factor in pairs, and in a minimal counterexample to the Herzog-Schönheim conjecture all exceed 2.

proposition_4_2: For pairwise coprime integers r1, r2, r3, no group has three pairwise disjoint cosets of subgroups of indices 2r1, 2r2 and 2r3.

proposition_4_3: For pairwise coprime integers r1, ..., r4, no group has four pairwise disjoint cosets of subgroups of indices 3r1, 3r2, 3r3 and 3r4.

proposition_4_5: For pairwise coprime integers r1, ..., r4 with r1 odd, no group has four pairwise disjoint cosets of subgroups of indices 2r1, 4r2, 4r3 and 4r4.

proposition_4_7: For pairwise coprime integers r2, ..., r5 with r2 odd, no group has five pairwise disjoint cosets of subgroups of indices 3, 3r2, 6r3, 6r4 and 6r5.

theorem_a: Every group of order less than 1440 satisfies the Herzog-Schönheim conjecture: no partition of it into two or more cosets has pairwise distinct indices.

theorem_b: For n at most 4, every n-tuple of indices of subgroups of a group G having pairwise disjoint cosets is also the tuple of moduli of pairwise disjoint arithmetic progressions.

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Margolis, Leo and Schnabel, Ofir, 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. The copy read for this card is arXiv:1803.03569v1 (9 March 2018, 16 pp.), the only arXiv version, whose labels the card uses. The arXiv record names arXiv's non-exclusive distribution license, every other right reserved.

The paper attacks the Herzog-Schönheim conjecture, the analogue for groups of the Davenport-Mirsky-Newman-Rado theorem (proving a conjecture of Erdős): when a group is partitioned into two or more cosets of subgroups of finite index, two of the cosets come from subgroups of the same index. Theorem A proves the conjecture for every group G of order less than 1440, extending the previous bound of 240 due to Ginosar. Theorem B, the key ingredient, proves that every G-harmonic n-tuple with n at most 4 is also Z-harmonic, answering a question of Ginosar in that range and generalizing a result of Zhu; here an n-tuple (a_1, ..., a_n) is called G-harmonic if G has subgroups of these indices with pairwise disjoint cosets, and Z-harmonic if pairwise disjoint arithmetic progressions with these differences exist. From Section 3 on the paper takes every group to be finite, so its propositions and proofs concern finite groups. The proof combines arithmetical restrictions on the indices of subgroups in a minimal counterexample (Section 2, using Egyptian-fraction relations and Lemma 2.2, which says U V = G implies (U, V) is not harmonic, so coprime indices are never G-harmonic) with a classification of the possible G-harmonic tuples for n at most 4; Theorem B reduces to excluding three tuple types, handled in Propositions 4.2, 4.3 and 4.5, and Theorem A additionally requires excluding the 5-tuples (3, 3r_2, 6r_3, 6r_4, 6r_5) with r_2, ..., r_5 pairwise coprime and r_2 odd (Proposition 4.7), a step its proof needs only for groups of order 1080. The paper bears on problem 274 by verifying the Herzog-Schönheim conjecture for every group of order below 1440.

Source: https://arxiv.org/abs/1803.03569.

Bears on.

• #274: Theorem A excludes, for every group of order below 1440, an exact covering by two or more cosets of pairwise different sizes; the problem for larger groups is not addressed. Propositions 4.2, 4.3, 4.5 and 4.7 are the four obstructions that the problem's Itabe claim page lists under Depends on.

Results. Labels and pages are those of arXiv:1803.03569v1.

• Theorem A (p. 1): every group of order less than 1440 satisfies the Herzog-Schönheim conjecture.
• Theorem B (p. 2): for $n\le4$, every $G$-harmonic $n$-tuple is $\mathbb Z$-harmonic.
• Lemma 2.3 (p. 3): the indices of a coset partition without multiplicity are pairwise distinct, have reciprocal sum 1 and pairwise common factors, and in a minimal counterexample exceed 2.
• Proposition 4.2 (p. 8): $(2r_1,2r_2,2r_3)$ with pairwise coprime $r_i$ is not $G$-harmonic.
• Proposition 4.3 (p. 8): $(3r_1,3r_2,3r_3,3r_4)$ with pairwise coprime $r_i$ is not $G$-harmonic.
• Proposition 4.5 (p. 10): $(2r_1,4r_2,4r_3,4r_4)$ with pairwise coprime $r_i$ and $r_1$ odd is not $G$-harmonic.
• Proposition 4.7 (p. 15): $(3,3r_2,6r_3,6r_4,6r_5)$ with pairwise coprime $r_i$ and $r_2$ odd is not $G$-harmonic; the proof of Theorem A uses it only for groups of order 1080.
• Lemma 2.2 (p. 3), which the paper cites from Ginosar and Schnabel (2011): if $UV=G$ then $(U,V)$ is not harmonic, so no pair of coprime integers is $G$-harmonic. It is not given a page; Lemma 2.3 and Theorem B record its use.

Read status. Claims checked: each linked statement was read clause by clause against the print; the proofs were read for their structure only, and nothing is independently reviewed.

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