# The twisted group ring isomorphism problem over fields

## TL;DR

This paper explores the twisted group ring isomorphism problem over fields, analyzing how twisted group rings encode group information, especially for abelian and non-abelian groups, and distinguishes certain groups with isomorphic group algebras.

## Contribution

It introduces a generalization of Schur covers applicable over non-algebraically closed fields and demonstrates how twisted group rings can differentiate groups with identical group algebras.

## Key findings

- Results depend on roots of unity in the field
- Generalized Schur covers exist over non-closed fields
- Twisted group rings distinguish certain non-isomorphic groups

## Abstract

Similarly to how the classical group ring isomorphism problem asks, for a commutative ring $R$, which information about a finite group $G$ is encoded in the group ring $RG$, the twisted group ring isomorphism problem asks which information about $G$ is encoded in all the twisted group rings of $G$ over $R$. We investigate this problem over fields. We start with abelian groups and show how the results depend on the roots of unity in $R$. In order to deal with non-abelian groups we construct a generalization of a Schur cover which exists also when $R$ is not an algebraically closed field, but still linearizes all projective representations of a group. We then show that groups from the celebrated example of Everett Dade which have isomorphic group algebras over any field can be distinguished by their twisted group algebras over finite fields.