Diameters of finite simple groups: sharp bounds and applications

• 成果类型: Article
• 署名作者: Liebeck, MW; Shalev, A
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/3062101
• 发表日期: 2001
• 页码: 383-406
• 关键词: probability; permutation; characters; number; POWERS

摘要

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 (x(1),. . .,x(d)). 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.