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 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.
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Acknowledgement
We thank Yuval Ginosar for useful discussions.
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The first author is a postdoctoral researcher of the Research Foundation Flanders (FWO—Vlaanderen). We are grateful for the Technion—Israel Institute of Technology, for supporting the first author’s visit to Haifa
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Margolis, L., Schnabel, O. The twisted group ring isomorphism problem over fields. Isr. J. Math. 238, 209–242 (2020). https://doi.org/10.1007/s11856-020-2017-9
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DOI: https://doi.org/10.1007/s11856-020-2017-9