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期刊论文
COVERING A FINITE ABELIAN GROUP BY SUBSET SUMS
Combinatorica 23 (4) (2003) 599-611,-0001,():
Let G be an abelian group of order n. The critical number c(G) of G is the smallests such that the subset sums set Σ(S) covers all G for eachs ubset S⊂G\{0} of cardinality |S|≥s. It has been recently proved that, if p is the smallest prime dividing n and n/p is composite, then c(G)=|G|/p+p−2, thus establishing a conjecture of Diderrich. We characterize the critical sets with |S|=|G|/p+p−3 and Σ(S)=G, wh ere p≥3 is the smallest prime dividing n, n/p is composite and n≥7p2+3p. We also extend a result of Diderrichan d Mann by proving that, for n≥67, |S|≥n/3+2 and S=G imply Σ(S)=G. Sets of cardinality |S|≥ n+11 4 for which Σ(S)=G are also characterized when n≥183, the smallest prime p dividing n is odd and n/p is composite. Finally we obtain a necessary and sufficient condition for the equality Σ(G)=G to hold when |S|≥n/(p+2)+p, wh ere p≥5, n/p is composite and n≥15p2.
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