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Twisted group ring isomorphism problem

Leo Margolis, Ofir Schnabel

TL;DR

This work introduces the twisted group ring isomorphism problem (TGRIP) for finite groups, focusing on complex twisted group algebras C^alpha G and the refined equivalence G ~_C H via a cohomology bijection and twisted isomorphisms. It develops a framework connecting ~_C to the ordinary group algebra, Schur multiplier M(G), and Schur covers S_G, and proves exact results for groups of order n with n = p^4 or n = p^2 q^2 under non-central-type hypotheses, including classifications within Omega_{p^4} and Omega_{p^2 q^2}. The paper also interrogates central-type groups, proving many are ~_C-singletons while providing a concrete central-type counterexample where two non-isomorphic groups are ~_C-equivalent, supported by computational data for order 64. Overall, it clarifies which classical invariants govern TGRIP and demonstrates both the power and limits of invariant-based classifications in twisted settings.

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.

Twisted group ring isomorphism problem

TL;DR

This work introduces the twisted group ring isomorphism problem (TGRIP) for finite groups, focusing on complex twisted group algebras C^alpha G and the refined equivalence G ~_C H via a cohomology bijection and twisted isomorphisms. It develops a framework connecting ~_C to the ordinary group algebra, Schur multiplier M(G), and Schur covers S_G, and proves exact results for groups of order n with n = p^4 or n = p^2 q^2 under non-central-type hypotheses, including classifications within Omega_{p^4} and Omega_{p^2 q^2}. The paper also interrogates central-type groups, proving many are ~_C-singletons while providing a concrete central-type counterexample where two non-isomorphic groups are ~_C-equivalent, supported by computational data for order 64. Overall, it clarifies which classical invariants govern TGRIP and demonstrates both the power and limits of invariant-based classifications in twisted settings.

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, -groups of order and groups of order , where and denote different primes.

Paper Structure

This paper contains 5 sections, 23 theorems, 59 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Useful results
  3. Examples and conditions
  4. Groups of order $p^4$ and $p^2q^2$
  5. Groups of central type

Key Result

Lemma 1.2

Any abelian group $A$ is a singleton with respect to $\sim_{{\mathbb C}}$.

Figures (1)

  • Figure 1: Some relations connected to the twisted group ring isomorphism problem

Theorems & Definitions (40)

  • Definition 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Theorem 2.4
  • ...and 30 more