JournalsjemsVol. 13, No. 4pp. 1063–1077

Product decompositions of quasirandom groups and a Jordan type theorem

  • Nikolay Nikolov

    University of Oxford, UK
  • László Pyber

    Hungarian Academy of Sciences, Budapest, Hungary
Product decompositions of quasirandom groups and a Jordan type theorem cover

Abstract

We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If kk is the minimal degree of a representation of the finite group~GG, then for every subset BB of GG with B>G/k13|B| > |G| / k^{\frac13} we have B3=GB^3 = 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 k2k \geq 2, then GG has a proper subgroup of index at most c0k2c_0 k^2 for some constant c0c_0, hence a product-free subset of size at least G/ck|G| / ck. This answers a question of Gowers.

Cite this article

Nikolay Nikolov, László Pyber, Product decompositions of quasirandom groups and a Jordan type theorem. J. Eur. Math. Soc. 13 (2011), no. 4, pp. 1063–1077

DOI 10.4171/JEMS/275