Twisted group ring isomorphism problem

TL;DR

This paper investigates a variation of the group ring isomorphism problem focusing on projective representations, introducing new conditions, methods, and results for specific classes of groups.

Contribution

It formulates a new variation of the classical isomorphism problem, explores logical relations among conditions, and provides results for various group classes.

Findings

Established conditions for abelian groups and groups of central type.

Analyzed p-groups of order p^4 and groups of order p^2q^2.

Developed methods to study the twisted group ring isomorphism problem.

Abstract

We propose and study a variation of the classical isomorphism problem for group rings in the context of projective representations. We formulate several weaker conditions following from our notion and give all logical connections between these condition by studying concrete examples. We introduce methods to study the problem and provide results for various classes of groups, including abelian groups, groups of central type, $p$-groups of order $p^4$ and groups of order $p^2q^2$, where $p$ and $q$ denote different primes.