The twisted group ring isomorphism problem over fields

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 characteristic of 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.

Publication year:2020

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