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11 August 2020
  » arxiv » math.GR/0306099

 Article overview

On the Herzog-Schönheim conjecture for uniform covers of groups
Zhi-Wei Sun ;
Date 5 Jun 2003
Journal J. Algebra 273(2004), no. 1, 153--175
Subject Group Theory; Number Theory MSC-class: 20D60; 05A18; 11B25; 11N45; 20D20; 20D35; 20E15; 20F16 | math.GR math.NT
AbstractLet G be any group and $a_1G_1,...,a_kG_k (k>1)$ 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 $n_1=[G:G_1],...,n_k=[G:G_k]$ cannot be distinct. In this paper we show that if $Cal A$ covers all the elements of G the same times and $G_1,...,G_k$ are subnormal subgroups of G not all equal to G, then $M=max_{1le jle k}|{1le ile k:n_i=n_j}|$ is not less than the smallest prime divisor of $n_1... n_k$, moreover $min_{1ls ils k}log n_i=O(Mlog^2 M)$ where the O-constant is absolute.
Source arXiv, math.GR/0306099
Other source [GID 140358] math.GR/0306099
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