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TwistedConjugacyClone

Computation with twisted conjugacy classes

3.0.0

21 July 2025

Sam Tertooy
Email: sam.tertooy@kuleuven.be
Homepage: https://stertooy.github.io/
Address:
Wiskunde
KU Leuven, Kulak Kortrijk Campus
Etienne Sabbelaan 53
8500 Kortrijk
Belgium

Abstract

The TwistedConjugacyClone package provides methods for solving the twisted conjugacy problem (including the "search" and "multiple" variants) and for computing Reidemeister classes, numbers, spectra, and zeta functions. It also includes utility functions for working with (double) cosets, group homomorphisms, and group derivations.

These methods are primarily designed for use with finite groups and with PcpGroups (finite or infinite) provided by the Polycyclic package.

Copyright

© 2020–2025 Sam Tertooy

The TwistedConjugacyClone package is free software, it may be redistributed and/or modified under the terms and conditions of the GNU Public License Version 2 or (at your option) any later version.

Acknowledgements

This documentation was created using the GAPDoc and AutoDoc packages.

Contents

1 The TwistedConjugacy package

1.1 Installation

2 Mathematical Background

3 Twisted conjugacy

3.1 The twisted conjugation action

3.1-1 TwistedConjugation

3.2 The twisted conjugacy (search) problem

3.2-1 IsTwistedConjugate
3.2-2 RepresentativeTwistedConjugation

3.3 The multiple twisted conjugacy (search) problem

4 Twisted conjugacy classes

4.1 Creating a twisted conjugacy class

4.1-1 TwistedConjugacyClass

4.2 Operations on twisted conjugacy classes

  4.2-1 Representative
  4.2-2 ActingDomain
  4.2-3 FunctionAction
  4.2-4 \in
  4.2-5 Size
  4.2-6 StabiliserOfExternalSet
  4.2-7 List
  4.2-8 Random
  4.2-9 \=

4.3 Calculating all twisted conjugacy classes

4.3-1 TwistedConjugacyClasses
4.3-2 RepresentativesTwistedConjugacyClasses

5 Reidemeister numbers and spectra

5.1 Reidemeister numbers

5.1-1 ReidemeisterNumber

5.2 Reidemeister spectra

  5.2-1 ReidemeisterSpectrum
  5.2-2 ExtendedReidemeisterSpectrum
  5.2-3 CoincidenceReidemeisterSpectrum
  5.2-4 TotalReidemeisterSpectrum

6 Reidemeister zeta functions

6.1 Reidemeister zeta functions

  6.1-1 ReidemeisterZetaCoefficients
  6.1-2 IsRationalReidemeisterZeta
  6.1-3 ReidemeisterZeta
  6.1-4 PrintReidemeisterZeta

7 Cosets of PcpGroups

7.1 Right cosets

7.1-1 Intersection

7.2 Double cosets

  7.2-1 \in
  7.2-2 Size
  7.2-3 List
  7.2-4 \=
  7.2-5 DoubleCosets
  7.2-6 DoubleCosetRepsAndSizes
  7.2-7 DoubleCosetIndex

8 Group homomorphisms

8.1 Representatives of homomorphisms between groups

  8.1-1 RepresentativesAutomorphismClasses
  8.1-2 RepresentativesEndomorphismClasses
  8.1-3 RepresentativesHomomorphismClasses

8.2 Coincidence and fixed point groups

8.2-1 FixedPointGroup
8.2-2 CoincidenceGroup

8.3 Induced and restricted group homomorphisms

8.3-1 InducedHomomorphism
8.3-2 RestrictedHomomorphism

9 Group derivations

9.1 Creating group derivations

9.1-1 GroupDerivationByImages
9.1-2 GroupDerivationByFunction

9.2 Operations for group derivations

  9.2-1 IsInjective
  9.2-2 IsSurjective
  9.2-3 IsBijective
  9.2-4 Kernel
  9.2-5 Image
  9.2-6 PreImagesRepresentative
  9.2-7 PreImagges

9.3 Images of group derivations

  9.3-1 \in
  9.3-2 Size
  9.3-3 List

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