Waring problem for finite quasisimple groups

Michael Larsen, Aner Shalev, Pham Huu Tiep

Research output: Contribution to journalArticle

8 Citations (Scopus)

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 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.

Original languageEnglish (US)
Pages (from-to)2323-2348
Number of pages26
JournalInternational Mathematics Research Notices
Volume2013
Issue number10
DOIs
StatePublished - 2013

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Waring's problem
Finite Group
Finite Simple Group
Alternating group
Simple group
Natural number
Lagrange
Cover
Analogue
Theorem

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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

In: International Mathematics Research Notices, Vol. 2013, No. 10, 2013, p. 2323-2348.

Research output: Contribution to journalArticle

Larsen, Michael ; Shalev, Aner ; Tiep, Pham Huu. / Waring problem for finite quasisimple groups. In: International Mathematics Research Notices. 2013 ; Vol. 2013, No. 10. pp. 2323-2348.
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