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 \((x,y) \mapsto x^Ny^N\) is surjective on every finite non-abelian simple group; if N is an odd integer, then the word map \((x,y,z) \mapsto x^Ny^Nz^N\) 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 \((x,y) \mapsto x^Ny^N\) that depend on the number of prime factors of the integer N.