Algorithms determining finite simple images of finitely presented groups

• 成果类型: Article
• 署名作者: Bridson, Martin R.; Evans, David M.; Liebeck, Martin W.; Segal, Dan
• 署名单位: University of Oxford; Imperial College London; University of Oxford
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-019-00892-3
• 发表日期: 2019
• 页码: 623-648
• 关键词: subgroup structure; algebraic-groups; definable sets

摘要

We address the question: for which collections of finite simple groups does there exist an algorithm that determines the images of an arbitrary finitely presented group that lie in the collection? We prove both positive and negative results. For a collection of finite simple groups that contains infinitely many alternating groups, or contains classical groups of unbounded dimensions, we prove that there is no such algorithm. On the other hand, for families of simple groups of Lie type of bounded rank, we obtain positive results. For example, for any fixed untwisted Lie type X there is an algorithm that determines whether or not any given finitely presented group has simple images of the form X(q) for infinitely many q, and if there are finitely many, the algorithm determines them.