Abstract
Let A be a finite subset of a group G 0 with |A −1 A|≤2|A−2. We show that there are an element α∈A and a non-null proper subgroup H of G such that one of the following holds:
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x −1 Hy⊂A −1 A, for all x,y∈A not both in Hα
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x Hy −1⊂AA −1, for all x,y∈A not both in αH
where G is the subgroup generated by A −1 A.
Assuming that A −1 A≠G and that \(\left| {A^{ - 1} A} \right| < \tfrac{{5|A|}} {3} \), we show that there are a normal subgroup K of G and a subgroup H with K⊂H⊂A −1 A and 2|K|≥|H| such that
.
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Yahya O. Hamidoune passed away on March 11, 2011. His friends and collaborators have helped in the final editorial work to honour his memory and have his latest results published.