On twisted group ring isomorphism problem for p-groups

TL;DR

This paper investigates the isomorphism problem for twisted complex group algebras of finite p-groups, introducing the generalized corank invariant and solving the problem for non-abelian p-groups with low corank.

Contribution

It introduces the generalized corank invariant and provides a solution for non-abelian p-groups with generalized corank up to three in the twisted group algebra isomorphism problem.

Findings

Solved the isomorphism problem for non-abelian p-groups with generalized corank ≤ 3.

Identified the role of the generalized corank as an invariant in twisted group algebra isomorphisms.

Extended understanding of the structure of twisted complex group algebras for specific p-groups.

Abstract

In this article, we explore the problem of determining isomorphisms between the twisted complex group algebras of finite $p$-groups. This problem bears similarity to the classical group algebra isomorphism problem and has been recently examined by Margolis-Schnabel. Our focus lies on a specific invariant, referred to as the generalized corank, which relates to the twisted complex group algebra isomorphism problem. We provide a solution for non-abelian $p$-groups with generalized corank at most three.