Geodesic
Product decompositions of quasirandom groups and a Jordan type theorem
Journal of the European Mathematical Society, Tome 13 (2011) no. 4, pp. 1063-1077.

Voir la notice de l'article provenant de la source EMS Press

We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If k is the minimal degree of a representation of the finite group G, then for every subset B of G with ∣B∣>∣G∣/k31​ we have B3=G. We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan's theorem which implies that if k≥2, then G has a proper subgroup of index at most c0​k2 for some constant c0​, hence a product-free subset of size at least ∣G∣/ck. This answers a question of Gowers.
DOI : 10.4171/jems/275
Classification : 20-XX, 00-XX
Keywords: Quasirandom groups, product-free sets, word values 
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Nikolay Nikolov; László Pyber. Product decompositions of quasirandom groups and a Jordan type theorem. Journal of the European Mathematical Society, Tome 13 (2011) no. 4, pp. 1063-1077. doi : 10.4171/jems/275. http://geodesic.mathdoc.fr/articles/10.4171/jems/275/

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