### Abstract

The classical Waring problem deals with expressing every natural number as a sum of g(k) kth powers. Similar problems for finite simple groups were studied recently, and in this paper we study them for finite quasisimple groups G. We show that for a fixed group word w≠1 and for G of sufficiently large order we have w(G)^{3}=G, namely every element of G is a product of three values of w. For various families of finite quasisimple groups, including covers of alternating groups, we obtain a stronger result, namely w(G)^{2}=G. However, in contrast with the case of simple groups studied in [14], we show that w(G)^{2}=G need not hold for all large G; moreover, if k>2, then x^{k}y^{k} is not surjective on infinitely many finite quasisimple groups. The case k=2 turns out to be exceptional. Indeed, our last result shows that every element of a finite quasisimple group is a product of two squares. This can be regarded as a noncommutative analog of Lagrange's four squares theorem.

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

Pages (from-to) | 2323-2348 |

Number of pages | 26 |

Journal | International Mathematics Research Notices |

Volume | 2013 |

Issue number | 10 |

DOIs | |

State | Published - 2013 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*International Mathematics Research Notices*,

*2013*(10), 2323-2348. https://doi.org/10.1093/imrn/rns109

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

Research output: Contribution to journal › Article

*International Mathematics Research Notices*, vol. 2013, no. 10, pp. 2323-2348. https://doi.org/10.1093/imrn/rns109

}

TY - JOUR

T1 - Waring problem for finite quasisimple groups

AU - Larsen, Michael

AU - Shalev, Aner

AU - Tiep, Pham Huu

PY - 2013

Y1 - 2013

N2 - The classical Waring problem deals with expressing every natural number as a sum of g(k) kth powers. Similar problems for finite simple groups were studied recently, and in this paper we study them for finite quasisimple groups G. We show that for a fixed group word w≠1 and for G of sufficiently large order we have w(G)3=G, namely every element of G is a product of three values of w. For various families of finite quasisimple groups, including covers of alternating groups, we obtain a stronger result, namely w(G)2=G. However, in contrast with the case of simple groups studied in [14], we show that w(G)2=G need not hold for all large G; moreover, if k>2, then xkyk is not surjective on infinitely many finite quasisimple groups. The case k=2 turns out to be exceptional. Indeed, our last result shows that every element of a finite quasisimple group is a product of two squares. This can be regarded as a noncommutative analog of Lagrange's four squares theorem.

AB - The classical Waring problem deals with expressing every natural number as a sum of g(k) kth powers. Similar problems for finite simple groups were studied recently, and in this paper we study them for finite quasisimple groups G. We show that for a fixed group word w≠1 and for G of sufficiently large order we have w(G)3=G, namely every element of G is a product of three values of w. For various families of finite quasisimple groups, including covers of alternating groups, we obtain a stronger result, namely w(G)2=G. However, in contrast with the case of simple groups studied in [14], we show that w(G)2=G need not hold for all large G; moreover, if k>2, then xkyk is not surjective on infinitely many finite quasisimple groups. The case k=2 turns out to be exceptional. Indeed, our last result shows that every element of a finite quasisimple group is a product of two squares. This can be regarded as a noncommutative analog of Lagrange's four squares theorem.

UR - http://www.scopus.com/inward/record.url?scp=84878065989&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84878065989&partnerID=8YFLogxK

U2 - 10.1093/imrn/rns109

DO - 10.1093/imrn/rns109

M3 - Article

VL - 2013

SP - 2323

EP - 2348

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 10

ER -