### Abstract

The classical Waring problem deals with expressing every natural num-ber as a sum of g(k) k-th powers. Recently there has been considerable interest in similar questions for non-abelian groups, and simple groups in particular. Here the k-th power word can be replaced by an arbitrary group word w ≠ 1, and the goal is to express group elements as short products of values of w. We give a best possible and somewhat surprising solution for this War-ing type problem for (non-abelian) finite simple groups of sufficiently high order, showing that a product of length two suffices to express all elements. Along the way we also obtain new results, possibly of independent in-terest, on character values in classical groups over finite fields, on regular semisimple elements lying in the image of word maps, and on products of conjugacy classes. Our methods involve algebraic geometry and representation theory, es-pecially Lusztig's theory of representations of groups of Lie type.

Original language | English (US) |
---|---|

Pages (from-to) | 1885-1950 |

Number of pages | 66 |

Journal | Annals of Mathematics |

Volume | 174 |

Issue number | 3 |

DOIs | |

State | Published - Nov 2011 |

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### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annals of Mathematics*,

*174*(3), 1885-1950. https://doi.org/10.4007/annals.2011.174.3.10

**The Waring problem for finite simple groups.** / Larsen, Michael; Shalev, Aner; Tiep, Pham Huu.

Research output: Contribution to journal › Article

*Annals of Mathematics*, vol. 174, no. 3, pp. 1885-1950. https://doi.org/10.4007/annals.2011.174.3.10

}

TY - JOUR

T1 - The Waring problem for finite simple groups

AU - Larsen, Michael

AU - Shalev, Aner

AU - Tiep, Pham Huu

PY - 2011/11

Y1 - 2011/11

N2 - The classical Waring problem deals with expressing every natural num-ber as a sum of g(k) k-th powers. Recently there has been considerable interest in similar questions for non-abelian groups, and simple groups in particular. Here the k-th power word can be replaced by an arbitrary group word w ≠ 1, and the goal is to express group elements as short products of values of w. We give a best possible and somewhat surprising solution for this War-ing type problem for (non-abelian) finite simple groups of sufficiently high order, showing that a product of length two suffices to express all elements. Along the way we also obtain new results, possibly of independent in-terest, on character values in classical groups over finite fields, on regular semisimple elements lying in the image of word maps, and on products of conjugacy classes. Our methods involve algebraic geometry and representation theory, es-pecially Lusztig's theory of representations of groups of Lie type.

AB - The classical Waring problem deals with expressing every natural num-ber as a sum of g(k) k-th powers. Recently there has been considerable interest in similar questions for non-abelian groups, and simple groups in particular. Here the k-th power word can be replaced by an arbitrary group word w ≠ 1, and the goal is to express group elements as short products of values of w. We give a best possible and somewhat surprising solution for this War-ing type problem for (non-abelian) finite simple groups of sufficiently high order, showing that a product of length two suffices to express all elements. Along the way we also obtain new results, possibly of independent in-terest, on character values in classical groups over finite fields, on regular semisimple elements lying in the image of word maps, and on products of conjugacy classes. Our methods involve algebraic geometry and representation theory, es-pecially Lusztig's theory of representations of groups of Lie type.

UR - http://www.scopus.com/inward/record.url?scp=80054990803&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80054990803&partnerID=8YFLogxK

U2 - 10.4007/annals.2011.174.3.10

DO - 10.4007/annals.2011.174.3.10

M3 - Article

VL - 174

SP - 1885

EP - 1950

JO - Annals of Mathematics

JF - Annals of Mathematics

SN - 0003-486X

IS - 3

ER -