Waring Problem for Finite Quasisimple Groups

TL;DR

This paper investigates the Waring problem for finite quasisimple groups, showing that for large groups, elements can be expressed as products of a few word values, with specific results for squares and higher powers.

Contribution

It extends the Waring problem to finite quasisimple groups, establishing new bounds on expressing elements as products of word values, including the case of squares.

Findings

w(G)^3=G for large G and fixed non-trivial word w

w(G)^2=G for certain families like covers of alternating groups

Every element is a product of two squares in finite quasisimple groups

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 have been studied recently, and in this paper we study them for finite quasisimple groups G. We show that for a fixed non-trivial group word w and large enough G we have w(G)^3=G, namely every element of G is a product of 3 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, we show that w(G)^2 need not equal G for all large G. If k>2 then x^k y^k fails to be surjective for 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 non-commutative analogue of Lagrange's four squares theorem.