Abstract
We prove that the Herzog–Schönheim Conjecture holds for any group G of order smaller than 1440. In other words we show that in any non-trivial coset partition \(\{g_i U_i\}_{i=1}^n \) of G there exist distinct \(1 \le i, j \le n\) such that \([G:U_i]=[G:U_j]\). We also study interaction between the indices of subgroups having cosets with pairwise trivial intersection and harmonic integers. We prove that if \(U_1,\ldots ,U_n\) are subgroups of G which have pairwise trivially intersecting cosets and \(n \le 4\) then \([G:U_1],\ldots ,[G:U_n]\) are harmonic integers.
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References
Berger, M.A., Felzenbaum, A., Fraenkel, A.: Remark on the multiplicity of a partition of a group into cosets. Fund. Math. 128(3), 139–144 (1987)
Chen, Y.-G.: A theorem on harmonic sequences. Discrete Math. 186(1–3), 287–288 (1998)
Ginosar, Y.: Tile the group. Elem. Math. 73(2), 66–73 (2018)
Ginosar, Y., Schnabel, O.: Prime factorization conditions providing multiplicities in coset partitions of groups. J. Comb. Number Theory 3(2), 75–86 (2011)
Huhn, A.P., Megyesi, L.: On disjoint residue classes. Discrete Math. 41(3), 327–330 (1982)
Herzog, M., Schönheim, J.: Research problem no. 9. Can. Math. Bull. 17, 150 (1974)
Huppert, B.: Endliche Gruppen. I. Die Grundlehren der Mathematischen Wissenschaften, vol. 134. Springer, Berlin (1967)
Korec, I., Znám, S.: On disjoint covering of groups by their cosets. Math. Slovaca 27(1), 3–7 (1977)
O’Bryant, K.: On Z.-W. Sun’s disjoint congruence classes conjecture, Combinatorial number theory, pp. 403–411. de Gruyter, Berlin (2007)
Porubský, S., Schönheim, J.: Covering systems of Paul Erdös. Past, present and future, Paul Erdös and his mathematics, I (Budapest, 1999), Bolyai Soc. Math. Stud., vol. 11, pp. 581–627. János Bolyai Math. Soc., Budapest (2002)
Sun, Z.-W.: Solutions to two problems of Huhn and Megyesi. Chin. Ann. Math. Ser. A 13(6), 722–727 (1992)
Sun, Z.-W.: Finite covers of groups by cosets or subgroups. Int. J. Math. 17(9), 1047–1064 (2006)
Zhu, W.-J.: On Sun’s conjecture concerning disjoint cosets. Int. J. Mod. Math. 3(2), 197–206 (2008)
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The first author has been supported by an individual Marie-Curie Individual Fellowship from H2020 EU-project 705112-ZC and the FWO (Research Foundation Flanders). The second author has been supported by the Minerva Stiftung and ISF Grant 797/14.
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Margolis, L., Schnabel, O. The Herzog–Schönheim conjecture for small groups and harmonic subgroups. Beitr Algebra Geom 60, 399–418 (2019). https://doi.org/10.1007/s13366-018-0419-1
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DOI: https://doi.org/10.1007/s13366-018-0419-1