Finite groups with the same character tables, Drinfel'd algebras and Galois algebras

TL;DR

This paper establishes a deep connection between finite groups with identical character tables and their associated algebraic structures, such as Drinfel'd quasi-bialgebras and Galois algebras, providing new algebraic characterizations.

Contribution

It demonstrates that finite groups sharing the same character tables correspond precisely to twisted forms of their group algebras as Drinfel'd quasi-bialgebras or non-associative bi-Galois algebras.

Findings

Finite groups with identical character tables are characterized by twisted Drinfel'd algebra structures.

Character-preserving automorphisms are interpreted via Drinfel'd algebra frameworks.

The paper links permutation representations with the algebraic structures of Drinfel'd algebras.

Abstract

We prove that finite groups have the same complex character tables iff the group algebras are twisted forms of each other as Drinfel'd quasi-bialgebras or iff there is non-associative bi-Galois algebra over these groups. The interpretations of class-preserving automorphisms and permutation representations with the same character in terms of Drinfel'd algebras are also given.