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Geometric approach to the modular isomorphism problem: groups of order 64

Leo Margolis, Taro Sakurai

Abstract

We introduce a procedure based on computational algebraic geometry to determine whether two algebras are isomorphic. We then apply it to show that if $R$ is a commutative unital ring in which $2$ is not invertible, $G$ is a group of order dividing $64$ and $H$ some group, then an isomorphism of unital algebras $RG \cong RH$ implies an isomorphism of groups $G \cong H$.

Geometric approach to the modular isomorphism problem: groups of order 64

Abstract

We introduce a procedure based on computational algebraic geometry to determine whether two algebras are isomorphic. We then apply it to show that if is a commutative unital ring in which is not invertible, is a group of order dividing and some group, then an isomorphism of unital algebras implies an isomorphism of groups .
Paper Structure (5 sections, 6 theorems, 20 equations, 4 tables)

This paper contains 5 sections, 6 theorems, 20 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Geometric approach
  3. Illustration: groups of order 8
  4. Groups of order 64
  5. Data for groups of order 64

Key Result

Theorem A

Let $R$ be a commutative unital ring in which $2$ is not invertible and let $G$ and $H$ be groups. If the order of $G$ divides $64$, then an isomorphism of group algebras $RG \cong RH$ as unital algebras implies $G \cong H$ as groups.

Theorems & Definitions (14)

  • Theorem A
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof : Proof of \ref{['prop:Minors']}
  • proof : Correctness of
  • proof : Correctness of
  • Corollary 2.4
  • Remark 3.1
  • ...and 4 more