PRODUCTS OF NORMAL SUBSETS

In this paper we consider which families of finite simple groups G have the property that for each ∊ > 0 there exists N > 0 such that, if |G| ≥ N and S, T are normal subsets of G with at least ∊|G| elements each, then every non-trivial element of G is the product of an element of S and an element of T. We show that this holds in a strong and effective sense for finite simple groups of Lie type of bounded rank, while it does not hold for alternating groups or groups of the form PSLn(q) where q is fixed and n → ∞. However, in the case S = T and G alternating this holds with an explicit bound on N in terms of ∊. Related problems and applications are also discussed. In particular we show that, if w1, w2 are non-trivial words, G is a finite simple group of Lie type of bounded rank, and for g ∈ G, P_w1(G_),w2(G₎(g) denotes the probability that g1g2 = g where gi ∈ wi(G) are chosen uniformly and independently, then, as |G| → ∞, the distribution P_w1(G_),w2(G₎ tends to the uniform distribution on G with respect to the L^∞ norm.