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.

中文翻译：

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域上的扭曲群环同构问题

类似于经典的群环同构问题，对于交换环 R ，关于有限群 G 的哪些信息编码在群环 RG 中，扭曲群环同构问题询问关于 G 的哪些信息编码在所有扭曲群中R 上的 G 环。我们在字段上调查这个问题。我们从阿贝尔群开始，并展示结果如何取决于 R 的特征。为了处理非阿贝尔群，我们构造了 Schur 覆盖的推广，当 R 不是代数闭域时也存在，但仍然线性化一个群的所有投影表示。然后，我们证明了来自著名例子 Everett Dade 在任何域上具有同构群代数的群可以通过它们在有限域上的扭曲群代数来区分。