On Covers of Abelian Groups by Cosets
نویسندگان
چکیده
Let G be any abelian group and {asGs}ks=1 be a finite system of cosets of subgroups G1, . . . , Gk. We show that if {asGs} k s=1 covers all the elements of G at least m times with the coset atGt irredundant then [G : Gt] 6 2 and furthermore k > m + f([G : Gt]), where f( ∏ r i=1 p αi i ) = ∑ r i=1 αi(pi − 1) if p1, . . . , pr are distinct primes and α1, . . . , αr 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.
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