论文标题
扭曲的组环同构问题和无限的同胞组
Twisted group ring isomorphism problem and infinite cohomology groups
论文作者
论文摘要
我们继续研究扭曲组代数的组环同构问题的变化。与以前的工作相反，我们包括不包含有限顺序的同伴的共同体学课程。这使我们能够在任何特征$ 0 $的领域中特别研究问题。我们证明，有限的$ g $和$ h $无法通过其理性的扭曲组代数来区分，而$ g $和$ h $可以通过其半简单扭曲的组代数来确定其他领域。这与以下事实相反：从$ g $的所有半简单组代数中获得的$ g $的结构信息已经在其合理的组代数中编码。 我们进一步表明，对于一个奇数$ p $，有订单$ p^4 $的组，这些订单$ p^4 $无法通过其扭曲的组代数来区分$ f $，对于任何特征的字段$ f $与$ p $不同。另一方面，我们证明，大肠杆菌（E. dade）构建的群体在任何领域都具有同构群代数，可以通过其理性扭曲的组代数来区分。我们还回答了一个关于足够条件的问题，即扭曲的组环同构问题可以解决复数。
We continue our investigation of a variation of the group ring isomorphism problem for twisted group algebras. Contrary to previous work, we include cohomology classes which do not contain any cocycle of finite order. This allows us to study the problem in particular over any field of characteristic $0$. We prove that there are finite groups $G$ and $H$ which can not be distinguished by their rational twisted group algebras, while $G$ and $H$ can be identified by their semi-simple twisted group algebras over other fields. This is in contrast with the fact that the structural information on $G$ which can obtained from all the semi-simple group algebras of $G$ is already encoded in its rational group algebra. We further show that for an odd prime $p$ there are groups of order $p^4$ which can not be distinguished by their twisted group algebras over $F$ for any field $F$ of characteristic different from $p$. On the other hand we prove that the groups constructed by E. Dade, which have isomorphic group algebras over any field, can be distinguished by their rational twisted group algebras. We also answer a question about sufficient conditions for the twisted group ring isomorphism problem to hold over the complex numbers.
