Waring Problem for Finite Quasisimple Groups

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^ky^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.