On the Twisted Group Ring Isomorphism Problem for a class of groups

TL;DR

This paper investigates the twisted group ring isomorphism problem for specific classes of groups, providing criteria, examples, and solutions for extra-special p-groups and groups of order p^5.

Contribution

It offers new criteria for the TGRIP based on quotients and solves the problem for certain classes of groups, including most groups of order p^5.

Findings

Criteria for TGRIP via quotient analysis

Solved TGRIP for extra-special p-groups

Solved TGRIP for groups of order p^5 (except five)

Abstract

The twisted group ring isomorphism problem (TGRIP) is a variation of the classical group ring isomorphism problem. It asks whether the ring structure of the twisted group ring determines the group up to isomorphism. In this article, we study the TGRIP for direct product and central product of groups. We provide some criteria to answer the TGRIP for groups by answering the TGRIP for the associated quotients. As an application of these results, we provide several examples. Finally, we answer the TGRIP for extra-special p-groups, and for the groups of order $p^5$, where $p \geq 5$ is a prime, except a list of five groups.