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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 in G, x of prime order, and |y| is divisible by at most two primes, such that the product of the conjugacy classes of x and y contain all non-central elements of 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 one of the main results of Larsen, Shalev and Tiep by showing that, given any positive integer m, if the order of a finite simple group S is at least f(m) for a specified function f, then every element in S is a product of two mth powers. Furthermore, the verbal width of the mth power word on any finite simple group S is at most g(m) for a specified function g. We also show that, given any two non-trivial words v, w, if G is a finite quasisimple group of large enough order, then v(G)w(G) contains all non-central elements of G.