On the modular isomorphism problem for groups of class 3
28th October 2020, 2:30 pm – 3:30 pm
Let G be a finite group and let R be a commutative ring. In 1940, G. Higman asked whether the isomorphism type of G is determined by its group ring RG. Although the Isomorphism Problem has generally a negative answer, the Modular Isomorphism Problem (MIP), for G a p-group and R a field of positive characteristic p, is still open. Examples of p-groups for which the (MIP) has a positive solution are abelian groups, groups of order dividing 2^9 or 3^7 or p^5, certain groups of maximal class, etc.
I will give an overview of the history of the (MIP) and will present recent joint work with Leo Margolis for groups of nilpotency class 3. In particular, our results yield new families of groups of order p^6 and p^7 for which the (MIP) has a positive solution.