TY - JOUR
T1 - The Herzog–Schönheim conjecture for small groups and harmonic subgroups
AU - Margolis, Leo
AU - Schnabel, Ofir
PY - 2019/9/1
Y1 - 2019/9/1
N2 - 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.
AB - 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.
KW - Herzog–Schönheim Conjecture
KW - Coset partitions
KW - Harmonic subgroups
UR - https://arxiv.org/abs/1803.03569
UR - http://www.scopus.com/inward/record.url?scp=85070010122&partnerID=8YFLogxK
U2 - 10.1007/s13366-018-0419-1
DO - 10.1007/s13366-018-0419-1
M3 - Article
SN - 0138-4821
VL - 60
SP - 399
EP - 418
JO - Beiträge zur Algebra und Geometrie
JF - Beiträge zur Algebra und Geometrie
IS - 3
ER -