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

Original language | English |
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Pages (from-to) | 589-695 |

Number of pages | 107 |

Journal | Inventiones Mathematicae |

Volume | 213 |

Issue number | 2 |

DOIs | |

State | Published - 1 Aug 2018 |

### Bibliographical note

Publisher Copyright:© 2018, Springer-Verlag GmbH Germany, part of Springer Nature.

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