Diameters of finite simple groups: sharp bounds and applications

  title={Diameters of finite simple groups: sharp bounds and applications},
  author={Martin W. Liebeck and Aner Shalev},
  journal={Annals of Mathematics},
Let G be a finite simple group and let S be a normal subset of G. We determine the diameter of the Cayley graph r(G, S) associated with G and S, up to a multiplicative constant. Many applications follow. For example, we deduce that there is a constant c such that every element of G is a product of c involutions (and we generalize this to elements of arbitrary order). We also show that for any word w = w(xl,..., xd), there is a constant c = c(w) such that for any simple group G on which w does… 
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