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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 | | Abstract: | Let 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 | | Services: | Forum | Review | PDF | Favorites | |
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