Abstract
LetG be an Abelian group written additively,B a finite subset ofG, and lett be a positive integer. Fort≦|B|, letB t denote the set of sums oft distinct elements overB. Furthermore, letK be a subgroup ofG and let σ denote the canonical homomorphism σ:G→G/K. WriteB t (modB t) forB tσ and writeB t (modK) forBσ. The following addition theorem in groups is proved. LetG be an Abelian group with no 2-torsion and letB a be finite subset ofG. Ift is a positive integer such thatt<|B| then |B t (modK)|≧|B (modK)| for any finite subgroupK ofG.
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References
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Diderrich, G.T. Sums of lengtht in Abelian groups. Israel J. Math. 14, 14–22 (1973). https://doi.org/10.1007/BF02761530
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DOI: https://doi.org/10.1007/BF02761530