Surjective word maps and Burnside’s p^aq^b theorem

We prove surjectivity of certain word maps on finite non-abelian simple groups. More precisely, we prove the following: if N is a product of two prime powers, then the word map (x, y) ↦ x^Ny^N is surjective on every finite non-abelian simple group; if N is an odd integer, then the word map (x, y, z) ↦ x^Ny^Nz^N is surjective on every finite quasisimple group. These generalize classical theorems of Burnside and Feit–Thompson. We also prove asymptotic results about the surjectivity of the word map (x, y) ↦ x^Ny^N that depend on the number of prime factors of the integer N.