B-theorem

Abstract: 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.