2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/156598Let 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 k\ge 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 a conjecture of Gao and Geroldinger. Our new method involves algebraic number theory and characters of abelian groups.10 pages, also related to Number Theory and CombinatoricsGroup TheoryNumber Theory20K99; 05D99, 05E99; 11B25; 11B75; 11R04; 11S99; 20C15; 20D60text