A negative solution to the Modular Isomorphism Problem

Abstract: Let $R$ be a ring. The Isomorphism Problem for group rings over $R$ asks whether the isomorphism type of a group $G$ is determined by the isomorphism type of the group ring $RG$. The special case where $R$ is a field with $p$ elements and $G$ is a finite $p$-group, for $p$ prime, is known as the Modular Isomorphism Problem.

The history of the Isomorphism Problem goes back to a seminal paper of G. Higman in the 1940s. The Modular Isomorphism Problem appeared in a survey paper by R. Brauer in 1963. While many relevant instances of the general Isomorphism Problem have been already resolved, the Modular Isomorphism Problem resisted until now.

In cooperation with Diego García and Leo Margolis we discovered recently two non-isomorphic groups of order $2^9$ whose group algebras over any field of characteristic $2$ are isomorphic. We will present this example and give an overview of the state of the art on the Isomorphism Problem.

group theory

Audience: researchers in the topic

GOThIC - Ischia Online Group Theory Conference

Series comments: Please send a message to andrea.caranti@unitn.it to receive a link to the Zoom room where the conference (a series of talks, actually) takes place.