  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 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.
