EFFECTIVE RESULTS ON THE WARING PROBLEM FOR FINITE SIMPLE GROUPS

Let G be a finite quasisimple group of Lie type. We show that there are regular semisimple elements x,y is an element of G, x of prime order, and vertical bar y vertical bar is divisible by at most two primes, such that x(G) . y(G) superset of G\Z(G). In fact in all but four cases, y can be chosen to be of square-free order. Using this result, we prove an effective version of a previous result of Larsen, Shalev, and Tiep by showing that, given any integer m >= 1, if the order of a finite simple group S is at least m(8m2), then every element in S is a product of two mth powers. Furthermore, the verbal width of en on any finite simple group S is at most 80m root 2log(2)m + 56. We also show that, given any two non-trivial words w(1), w(2), if G is a finite quasisimple group of large enough order, then w(1) (G)w(2)(G) superset of G\Z (G).