Diameters of finite simple groups: Sharp bounds and applications

Martin W. Liebeck, Aner Shalev*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

98 Scopus citations

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 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.

Original languageAmerican English
Pages (from-to)383-406
Number of pages24
JournalAnnals of Mathematics
Volume154
Issue number2
DOIs
StatePublished - Sep 2001

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