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Diameters of Finite Simple Groups: Sharp Bounds and Applications

Martin W. Liebeck and Aner Shalev
Annals of Mathematics
Second Series, Vol. 154, No. 2 (Sep., 2001), pp. 383-406
Published by: Annals of Mathematics
DOI: 10.2307/3062101
Stable URL: http://www.jstor.org/stable/3062101
Page Count: 24
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Diameters of Finite Simple Groups: Sharp Bounds and Applications
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Abstract

Let G be a finite simple group and let S be a normal subset of G. We determine the diameter of the Cayley graph Γ(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(x1,..., xd), there is a constant c = c(w) such that for any simple group G on which w does not vanish, every element of G is a product of c values of w. From this we deduce that every verbal subgroup of a semisimple profinite group is closed. Other applications concern covering numbers, expanders, and random walks on finite simple groups.

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