Two inverse results


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 αA and a non-null proper subgroup H of G such that one of the following holds:

  • x −1 HyA −1 A, for all x,yA not both in

  • x Hy −1AA −1, for all x,yA not both in αH

where G is the subgroup generated by A −1 A.

Assuming that A −1 AG 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 KHA −1 A and 2|K|≥|H| such that

$A^{ - 1} AK = KA^{ - 1} A = A^{ - 1} Aand6|K| \geqslant |A^{ - 1} A| = 3|H|$


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Correspondence to Yahya Ould Hamidoune.

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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.

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Hamidoune, Y.O. Two inverse results. Combinatorica 33, 217–230 (2013).

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Mathematics Subject Classification (2000)

  • 11B60
  • 11B34
  • 20D60