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The Herzog–Schönheim conjecture for small groups and harmonic subgroups

Journal Contribution - Journal Article

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 {giUi}i=1n of G there exist distinct 1 ≤ i, j≤ 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, … , U n are subgroups of G which have pairwise trivially intersecting cosets and n≤ 4 then [G: U 1] , … , [G: U n] are harmonic integers.

Journal: Contributions to Algebra and Geometry
ISSN: 0138-4821
Issue: 3
Volume: 60
Pages: 399-418
Keywords:Herzog–Schönheim Conjecture, Coset partitions, Harmonic subgroups
Accessibility:Closed