TY - JOUR

T1 - Diameters of finite simple groups

T2 - Sharp bounds and applications

AU - Liebeck, Martin W.

AU - Shalev, Aner

PY - 2001/9

Y1 - 2001/9

N2 - Let G be a finite simple group and let S be a normal subset of G. We determine the diameter of the Cayley graph F(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(χ1,...,χd), 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.

AB - Let G be a finite simple group and let S be a normal subset of G. We determine the diameter of the Cayley graph F(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(χ1,...,χd), 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.

UR - http://www.scopus.com/inward/record.url?scp=0035540759&partnerID=8YFLogxK

U2 - 10.2307/3062101

DO - 10.2307/3062101

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AN - SCOPUS:0035540759

SN - 0003-486X

VL - 154

SP - 383

EP - 406

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 2

ER -