报告主题：A criterion for the modular isomorphism problem
报告人：DR.Taro Sakurai （Chiba University, Japan）
报告摘要：A fundamental theme in the study of group algebras is that what kind of properties of underlying groups (or even groups themselves) can be detected from the structure of algebras. The modular isomorphism problem—which is open for more than 60 years—asks whether an isomorphism GF(p)G = GF(p)H as unital GF(p)-algebras implies an isomorphism G = H as groups for finite p-groups G and H. Despite the problem can be stated in fairly simple terminology, this fundamental problem is considered to be very hard; even the case for finite p-groups of nilpotency class two is not solved. In this talk, we introduce a new class of finite groups (hereditary groups) and provide a new criterion (sufficient condition) for the problem. The proof rests on adjoin and counting homomorphisms, which implies it is more combinatorial and indirect approach rather than algebraic and direct approach in nature. New proofs for the classic theorems by Deskins and Passi-Sehgal which yield positive answer to the problem for small nilpotency class and exponent are provided with the help of theorems of Ault-Watters and Bovdi on the realizability of finite p-groups as quasi-regular groups of finite quasi-regular algebras.