The twisted group ring isomorphism problem over fields

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.