Waring Problem for Finite Quasisimple Groups

Author： Larsen Michael Shalev Aner Tiep Pham Huu

ISSN： 1073-7928

Source： International Mathematics Research Notices, Vol.2013, Iss.10, 2013-04, pp. : 2323-2348

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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 w1 and for g of sufficiently large order we have w(g)³g, 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)²g. However, in contrast with the case of simple groups studied in [14], we show that w(g)²g need not hold for all large g; moreover, if k>2, then x^ky^k is not surjective on infinitely many finite quasisimple groups. The case k2 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 Lagranges four squares theorem.