Twisted group ring isomorphism problem and infinite cohomology groups

TL;DR

This paper explores the twisted group ring isomorphism problem, revealing that certain finite groups cannot be distinguished by their twisted group algebras over various fields, and extends understanding of cohomology classes in this context.

Contribution

It introduces the inclusion of non-finite order cohomology classes in the study of twisted group algebras, expanding analysis over fields of characteristic zero and distinguishing groups with identical classical invariants.

Findings

Finite groups indistinguishable by rational twisted group algebras

Groups of order p^4 indistinguishable over fields of characteristic ≠ p

Dade's groups distinguished by rational twisted group algebras

Abstract

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.