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ModIsom

Computing automorphisms and checking isomorphisms for modular group algebras of finite p-groups

3.2.0

6 October 2026

Bettina Eick
Email: beick@tu-bs.de
Homepage: http://www.iaa.tu-bs.de/beick
Address:
Institut Analysis und Algebra
TU Braunschweig
Universitätsplatz 2
D-38106 Braunschweig
Germany

Diego Garcia-Lucas
Email: diego.garcial@usc.es
Address:
Departamento de Matematicas
Facultad de Matematicas
Universidad de Santiago de Compostela
Rua de Lope Gomez de Marzoa, s/nES-15705 Santiago de Compostela
Spain

Leo Margolis
Email: leo.margolis@uam.es
Homepage: http://www.margollo.github.io
Address:
Departamento de Matematicas
Universidad Autonoma de Madrid
Campus Cantoblanco
28049 Madrid
Spain

Tobias Moede
Email: t.moede@tu-braunschweig.de
Homepage: https://www.tu-braunschweig.de/iaa/personal/moede
Address:
Institute of Analysis and Algebra
TU Braunschweig
Universitaetsplatz 2, 38106 Braunschweig
Germany

Contents

1.1 Associative algebras and nilpotency

1.2 Isomorphisms and Automorphisms

1.3 The Modular Isomorphism Problem (MIP)

1.4 A nilpotent quotient algorithm

1.5 Kurosh Algebras

2.1 Nilpotent tables

  2.1-1 GetEntryTable
  2.1-2 MultByTable
  2.1-3 CompareTables
  2.1-4 CheckAssociativity
  2.1-5 CheckCommutativity
  2.1-6 CheckConsistency

2.2 Algebras in the GAP sense

2.2-1 AlgebraByTable
2.2-2 NilpotentTableOfRad

2.3 Tables for the Modular Isomorphism Problem

  2.3-1 TableOfRadQuotient
  2.3-2 ModIsomTable
  2.3-3 MIPElementTableToAlgebra
  2.3-4 MIPElementAlgebraToTable

3 Automorphism groups and Canonical Forms

3.1 Automorphism groups

3.1-1 AutGroupOfTable

3.2 Canonical forms

3.2-1 CanonicalFormOfTable
3.2-2 CanoFormWithAutGroupOfTable

3.3 Example of canonical form computation

4 The modular isomorphism problem

4.1 Computing bins and checking bins

  4.1-1 BinsByGT
  4.1-2 MIPSplitGroupsByGroupTheoreticalInvariants
  4.1-3 MIPSplitGroupsByGroupTheoreticalInvariantsNoCohomology
  4.1-4 BinsByGTAllFields
  4.1-5 MIPSplitGroupsByAlgebras
  4.1-6 MIPBinSplit

4.2 Kernel size

4.2-1 KernelSizePowerMap

4.3 The group theoretical invariants

  4.3-1 GroupInfo
  4.3-2 CenterDerivedInfo
  4.3-3 SandlingInfo
  4.3-4 JenningsInfo
  4.3-5 JenningsDerivedInfo
  4.3-6 BaginskiInfo
  4.3-7 BaginskiCarantiInfo
  4.3-8 NilpotencyClassInfo
  4.3-9 Theorem41MS22
  4.3-10 CyclicDerivedInfo
  4.3-11 MaximalAbelianDirectFactor
  4.3-12 SuccessorsN
  4.3-13 CanonicalNormalSubgroups
  4.3-14 NormalSubgroupsInfo
  4.3-15 IsCoveredByTheory
  4.3-16 DimensionTwoCohomology
  4.3-17 ConjugacyClassInfo
  4.3-18 SubgroupsInfo

5 Nilpotent Quotients

5.1 Computing nilpotent quotients

5.1-1 NilpotentQuotientOfFpAlgebra

5.2 Example of nilpotent quotient computation

6 Relatively free Algebras

6.1 Computing Kurosh Algebras

6.1-1 KuroshAlgebra
6.1-2 ExpandExponentLaw

6.2 A Library of Kurosh Algebras

6.2-1 KuroshAlgebraByLib

6.3 Example of accessing the library of Kurosh algebras

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