2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/132123We 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 $|B| > |G| / k^{1/3}$ we have B^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 k>1, then G has a proper subgroup of index at most ck^2 for some absolute constant c, hence a product-free subset of size at least $|G| / c'k$. This answers a question of Gowers.18 pages. In this third version we added an Appendix with a short proof of Proposition 0Group TheoryCombinatorics20D06; 20F69text