Surjective word maps and Burnside's theorem

• 成果类型: Article
• 署名作者: Guralnick, Robert M.; Liebeck, Martin W.; O'Brien, E. A.; Shalev, Aner; Tiep, Pham Huu
• 署名单位: University of Southern California; Imperial College London; University of Auckland; Hebrew University of Jerusalem; Rutgers University System; Rutgers University New Brunswick
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-018-0795-z
• 发表日期: 2018
• 页码: 589-695
• 关键词: finite simple-groups; conjugacy classes; unipotent characters; exceptional groups; waring problem; sharp bounds; REPRESENTATIONS; PRODUCTS; SUBGROUPS; GROWTH

摘要

We prove surjectivity of certain word maps on finite non-abelian simple groups. More precisely, we prove the following: if N is a product of two prime powers, then the word map is surjective on every finite non-abelian simple group; if N is an odd integer, then the word map is surjective on every finite quasisimple group. These generalize classical theorems of Burnside and Feit-Thompson. We also prove asymptotic results about the surjectivity of the word map that depend on the number of prime factors of the integer N.