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11 August 2020
 
  » arxiv » math.GR/0411144

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On Covers of Abelian Groups by Cosets
Günter Lettl ; Zhi-Wei Sun ;
Date 7 Nov 2004
Subject Group Theory; Number Theory MSC-class: 20D60; 05A05; 11B25; 11B75; 11R04; 20C15 | math.GR math.NT
AbstractLet G be any abelian group and {a_sG_s}_{s=1}^k be a finite system of cosets of subgroups G_1,...,G_k. We show that if {a_sG_s}_{s=1}^k covers all the elements of G at least m times with the coset a_tG_t irredundant then [G:G_t]le 2^{k-m} and furthermore kge m+f([G:G_t]), where f(prod_{i=1}^r p_i^{alpha_i})=sum_{i=1}^r alpha_i(p_i-1) if p_1,...,p_r are distinct primes and alpha_1,...,alpha_r are nonnegative integers. This extends Mycielski’s conjecture in a new way and implies an open conjecture of Gao and Geroldinger. Our new method involves algebraic number theory and characters of abelian groups.
Source arXiv, math.GR/0411144
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