本视频为群与图报告系列 Seminars on Groups and Graphs (https://www.gtseminar.xyz)。本次报告的ppt也在这里。
1987年，菲尔兹奖得主John G. Thompson在给中国群论学者施武杰教授的信中提出关于“同阶型群”可解性的重要问题。在2024年，牛津大学博士生Paweł Piwek给出了该问题的首个反例。本视频为Paweł Piwek对于此问题的学术报告。提出问题时，Thompson曾在信中评论：“The problem arose initially in the study of algebraic number fields, and is of considerable interest.”
标题: Solvability and Order Type for Finite Groups
摘要: How much can the order type — the list of element orders (with multiplicities) — reveal about the structure of a finite group G? Can it tell us whether G is abelian, nilpotent? Can it always determine whether G is solvable? This last question was posed in 1987 by John G. Thompson and I answered it negatively last year. The search for a counterexample was quite a puzzle hunt! It involved turning the problem into linear algebra and solving an integer matrix equation Ax=b. This would be easy if not for the fact that the size of A was 100,000 by 10,000…
About the speaker: Paweł Piwek obtained his PhD at the University of Oxford under the supervision of Professor Martin Bridson, a leading figure in Geometric Group Theory. His research focuses on profinite rigidity—distinguishing groups by their finite quotients—and group extensions, with findings published in several high-quality journals. He won medals in the International Mathematical Olympiad (IMO) in 2015 and 2016 and co-founded Maths Beyond Limits, an international camp that helps mathematically talented high school students connect and grow. Paweł is now a quantitative researcher at Optiver, gaining experience in programming and quantitative finance.
