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By now, we have a product theorem in every finite simple group $G$ of Lietype, with the strength of the bound depending only in the rank of $G$. Suchtheorems have numerous consequences: bounds on the diameters of Cayley graphs,spectral gaps, and so forth. For the alternating group Alt_n, we have aquasipolylogarithmic diameter bound (Helfgott-Seress 2014), but it does notrest on a product theorem.  We shall revisit the proof of the bound for Alt_n, bringing it closer to theproof for linear algebraic groups, and making some common themes clearer. As aresult, we will show how to prove a product theorem for Alt_n -- not of fullstrength, as that would be impossible, but strong enough to imply the diameterbound.

Growth in linear algebraic groups and permutation groups: towards a unified perspective