  
  
                               [1X TwistedConjugacy [101X
  
  
                  [1X Computation with twisted conjugacy classes [101X
  
  
                                     3.1.1
  
  
                                12 October 2025
  
  
                                  Sam Tertooy
  
  
  
  Sam Tertooy
      Email:    [7Xmailto:sam.tertooy@kuleuven.be[107X
      Homepage: [7Xhttps://stertooy.github.io/[107X
      Address:  [33X[0;14YWiskunde[133X
                [33X[0;14YKU Leuven, Kulak Kortrijk Campus[133X
                [33X[0;14YEtienne Sabbelaan 53[133X
                [33X[0;14Y8500 Kortrijk[133X
                [33X[0;14YBelgium[133X
  
  
  
  -------------------------------------------------------
  [1XAbstract[101X
  [33X[0;0YThe  [5XTwistedConjugacy[105X  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.[133X
  
  [33X[0;0YThese  methods  are  primarily  designed for use with finite groups and with
  PcpGroups (finite or infinite) provided by the [5XPolycyclic[105X package.[133X
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y© 2020–2025 Sam Tertooy[133X
  
  [33X[0;0YThe  [5XTwistedConjugacy[105X  package  is  free  software,  it may be redistributed
  and/or  modified  under  the  terms and conditions of the GNU Public License
  Version 2 ([7Xhttps://www.gnu.org/licenses/old-licenses/gpl-2.0.en.html[107X) or (at
  your option) any later version.[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YThis documentation was created using the [5XGAPDoc[105X and [5XAutoDoc[105X packages.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (TwistedConjugacy)[101X
  
  1 [33X[0;0YThe TwistedConjugacy package[133X
    1.1 [33X[0;0YInstallation[133X
    1.2 [33X[0;0YLoading[133X
    1.3 [33X[0;0YCiting[133X
    1.4 [33X[0;0YSupport[133X
  2 [33X[0;0YMathematical background[133X
  3 [33X[0;0YTwisted conjugacy[133X
    3.1 [33X[0;0YThe twisted conjugation action[133X
      3.1-1 TwistedConjugation
    3.2 [33X[0;0YThe twisted conjugacy (search) problem[133X
      3.2-1 IsTwistedConjugate
      3.2-2 RepresentativeTwistedConjugation
    3.3 [33X[0;0YThe multiple twisted conjugacy (search) problem[133X
  4 [33X[0;0YTwisted conjugacy classes[133X
    4.1 [33X[0;0YCreating a twisted conjugacy class[133X
      4.1-1 TwistedConjugacyClass
    4.2 [33X[0;0YOperations on twisted conjugacy classes[133X
      4.2-1 [33X[0;0YRepresentative[133X
      4.2-2 [33X[0;0YActingDomain[133X
      4.2-3 [33X[0;0YFunctionAction[133X
      4.2-4 [33X[0;0Y\in[133X
      4.2-5 [33X[0;0YSize[133X
      4.2-6 [33X[0;0YStabiliserOfExternalSet[133X
      4.2-7 [33X[0;0YList[133X
      4.2-8 [33X[0;0YRandom[133X
      4.2-9 [33X[0;0Y\=[133X
    4.3 [33X[0;0YCalculating all twisted conjugacy classes[133X
      4.3-1 TwistedConjugacyClasses
      4.3-2 RepresentativesTwistedConjugacyClasses
  5 [33X[0;0YReidemeister numbers and spectra[133X
    5.1 [33X[0;0YReidemeister numbers[133X
      5.1-1 ReidemeisterNumber
    5.2 [33X[0;0YReidemeister spectra[133X
      5.2-1 ReidemeisterSpectrum
      5.2-2 ExtendedReidemeisterSpectrum
      5.2-3 CoincidenceReidemeisterSpectrum
      5.2-4 TotalReidemeisterSpectrum
  6 [33X[0;0YReidemeister zeta functions[133X
    6.1 [33X[0;0YReidemeister zeta functions[133X
      6.1-1 ReidemeisterZetaCoefficients
      6.1-2 IsRationalReidemeisterZeta
      6.1-3 ReidemeisterZeta
      6.1-4 PrintReidemeisterZeta
  7 [33X[0;0YCosets of PcpGroups[133X
    7.1 [33X[0;0YRight cosets[133X
      7.1-1 Intersection
    7.2 [33X[0;0YDouble cosets[133X
      7.2-1 [33X[0;0Y\in[133X
      7.2-2 [33X[0;0YSize[133X
      7.2-3 [33X[0;0YList[133X
      7.2-4 [33X[0;0Y\=[133X
      7.2-5 [33X[0;0YDoubleCosets[133X
      7.2-6 [33X[0;0YDoubleCosetRepsAndSizes[133X
      7.2-7 [33X[0;0YDoubleCosetIndex[133X
  8 [33X[0;0YGroup homomorphisms[133X
    8.1 [33X[0;0YRepresentatives of homomorphisms between groups[133X
      8.1-1 RepresentativesAutomorphismClasses
      8.1-2 RepresentativesEndomorphismClasses
      8.1-3 RepresentativesHomomorphismClasses
    8.2 [33X[0;0YCoincidence and fixed point groups[133X
      8.2-1 FixedPointGroup
      8.2-2 CoincidenceGroup
    8.3 [33X[0;0YInduced and restricted group homomorphisms[133X
      8.3-1 InducedHomomorphism
      8.3-2 RestrictedHomomorphism
  9 [33X[0;0YGroup derivations[133X
    9.1 [33X[0;0YCreating group derivations[133X
      9.1-1 GroupDerivationByImages
      9.1-2 GroupDerivationByFunction
      9.1-3 GroupDerivationByAffineAction
    9.2 [33X[0;0YOperations for group derivations[133X
      9.2-1 [33X[0;0YIsInjective[133X
      9.2-2 [33X[0;0YIsSurjective[133X
      9.2-3 [33X[0;0YIsBijective[133X
      9.2-4 [33X[0;0YKernel[133X
      9.2-5 [33X[0;0YImage[133X
      9.2-6 [33X[0;0YPreImagesRepresentative[133X
      9.2-7 [33X[0;0YPreImages[133X
    9.3 [33X[0;0YImages of group derivations[133X
      9.3-1 [33X[0;0Y\in[133X
      9.3-2 [33X[0;0YSize[133X
      9.3-3 [33X[0;0YList[133X
  10 [33X[0;0YAffine actions[133X
    10.1 [33X[0;0YCreating an affine action[133X
      10.1-1 AffineActionByGroupDerivation
    10.2 [33X[0;0YOperations for affine actions[133X
      10.2-1 [33X[0;0YOrbitAffineAction[133X
      10.2-2 [33X[0;0YOrbitsAffineAction[133X
      10.2-3 [33X[0;0YNrOrbitsAffineAction[133X
      10.2-4 [33X[0;0YStabiliserAffineAction[133X
      10.2-5 [33X[0;0YRepresentativeAffineAction[133X
    10.3 [33X[0;0YOperations on orbits of affine actions[133X
      10.3-1 [33X[0;0YRepresentative[133X
      10.3-2 [33X[0;0YActingDomain[133X
      10.3-3 [33X[0;0YFunctionAction[133X
      10.3-4 [33X[0;0Y\in[133X
      10.3-5 [33X[0;0YSize[133X
      10.3-6 [33X[0;0YStabiliserOfExternalSet[133X
      10.3-7 [33X[0;0YList[133X
      10.3-8 [33X[0;0YRandom[133X
      10.3-9 [33X[0;0Y\=[133X
  
  
  [32X
