On the Herzog-Schönheim conjecture for uniform covers of groups

2003/06/30 by Zhi-Wei Sun
Mathematics · #math.GR #math.NT #msc:20D60 #msc:05A18 #msc:11B25 #msc:11N45 #msc:20D20 #msc:20D35 #msc:20E15 #msc:20F16

paper · pdf

published as J. Algebra 273(2004), no. 1, 153--175 · 22 pages

arxiv created 2004/12/30 · arxiv updated 2009/11/30

Abstract

Let G be any group and a₁G₁,...,a_kG_k (k>1) be left cosets in G. In 1974 Herzog and Schönheim conjectured that if \Cal A=\a_iG_i\_i=1^k is a partition of G then the (finite) indices n₁=[G:G₁],...,n_k=[G:G_k] cannot be distinct. In this paper we show that if \Cal A covers all the elements of G the same times and G₁,...,G_k are subnormal subgroups of G not all equal to G, then M=max_{1≤ j≤ k}|\1≤ i≤ k:n_i=n_j\| is not less than the smallest prime divisor of n₁... n_k, moreover min_{1\ls i\ls k}log n_i=O(Mlog² M) where the O-constant is absolute.

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• The space of coset partitions of F_n and Herzog-Schönheim conjecture
• The Herzog-Schönheim Conjecture for small groups and harmonic subgroups
• The Herzog-Schönheim conjecture for simple and symmetric groups
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