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The small sumsets property for solvable finite groups

Published in European Journal of Combinatorics. 2006, vol. 27, no. 7, p. 1102 - 1110
Abstract Let G be a group written multiplicatively.We say that G has the small sumsets property if for all positive integers r, s ≤ |G|, there exist subsets A, B ⊂ G such that |A| = r, |B| = s and |A · B| ≤ r + s − 1. If, in addition, it is possible to simultaneously satisfy A ⊂ B whenever r ≤ s, we speak of the nested small sumsets property for G. We prove that finite solvable groups satisfy this stronger form of the property. In the finite non-solvable case, we prove that subsets A, B ⊂ G satisfying |A| = r, |B| = s and |A· B| ≤ r +s−1 also exist, provided either r ≤ 12 or r + s ≥ |G| − 11.
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ELIAHOU, Shalom, KERVAIRE, Michel. The small sumsets property for solvable finite groups. In: European Journal of Combinatorics, 2006, vol. 27, n° 7, p. 1102 - 1110. doi: 10.1016/j.ejc.2006.06.004 https://archive-ouverte.unige.ch/unige:12107

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Deposited on : 2010-10-15

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