  
  
                                     [1X HeLP [101X
  
  
                        [1X Hertweck-Luthar-Passi method. [101X
  
  
                                      3.5
  
  
                                20 December 2019
  
  
                                 Andreas Bächle
  
                                  Leo Margolis
  
  
  
  Andreas Bächle
      Email:    [7Xmailto:ABachle@vub.ac.be[107X
      Homepage: [7Xhttp://homepages.vub.ac.be/~abachle/[107X
      Address:  [33X[0;14YVrije Universiteit Brussel[133X
                [33X[0;14YVakgroep Wiskunde[133X
                [33X[0;14YPleinlaan 2[133X
                [33X[0;14Y1050 Brussels[133X
                [33X[0;14YBelgium[133X
  
  
  Leo Margolis
      Email:    [7Xmailto:Leo.Margolis@vub.be[107X
      Homepage: [7Xhttp://homepages.vub.ac.be/~lmargoli/[107X
      Address:  [33X[0;14YVrije Universiteit Brussel[133X
                [33X[0;14YVakgroep Wiskunde[133X
                [33X[0;14YPleinlaan 2[133X
                [33X[0;14Y1050 Brussels[133X
                [33X[0;14YBelgium[133X
  
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y© 2017 by Andreas Bächle and Leo Margolis[133X
  
  [33X[0;0YThis  package  is  free  software and may be distributed under the terms and
  conditions  of  the  GNU  Public  License Version 2, or (at your option) any
  later version.[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YThe  authors  are grateful to Sebastian Gutsche, Christof Söger and Max Horn
  for  endowing  GAP  with  a 4ti2-Interface and a normaliz-Interface. We also
  would  like to thank Gutsche and Söger for many very helpful discussions. We
  also  want  to  give  credits  to  the  developers of the softwares 4ti2 and
  normaliz.  Thanks  go  to  David  Avis  for writing lrslib and answering our
  questions  about  it. We moreover thank Wolfgang Kimmerle for introducing us
  to  the  beautiful world of group rings. The development of this package was
  partially  supported  by the Research Foundation Flanders (FWO - Vlaanderen)
  and  the  DFG priority program SPP 1489 Algorithmic and Experimental Methods
  in Algebra, Geometry, and Number Theory.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (HeLP)[101X
  
  1 [33X[0;0YIntroduction[133X
  2 [33X[0;0YThe main functions[133X
    2.1 [33X[0;0YZassenhaus Conjecture[133X
      2.1-1 HeLP_ZC
    2.2 [33X[0;0YPrime Graph Question[133X
      2.2-1 HeLP_PQ
  3 [33X[0;0YFurther functions[133X
    3.1 [33X[0;0YChecks for specific orders[133X
      3.1-1 HeLP_WithGivenOrder
      3.1-2 HeLP_WithGivenOrderAndPA
      3.1-3 HeLP_WithGivenOrderAllTables
      3.1-4 HeLP_WithGivenOrderAndPAAllTables
      3.1-5 HeLP_WithGivenOrderAndPAAndSpecificSystem
    3.2 [33X[0;0YChecks for specific orders with s-constant characters[133X
      3.2-1 HeLP_WithGivenOrderSConstant
      3.2-2 HeLP_AddGaloisCharacterSums
    3.3 [33X[0;0YChecks for all orders[133X
      3.3-1 HeLP_AllOrders
      3.3-2 HeLP_AllOrdersPQ
    3.4 [33X[0;0YChanging the used Character Table[133X
      3.4-1 HeLP_ChangeCharKeepSols
      3.4-2 HeLP_Reset
    3.5 [33X[0;0YInfluencing how the Systems of Inequalities are solved[133X
      3.5-1 HeLP_Solver
      3.5-2 HeLP_UseRedund
      3.5-3 HeLP_Change4ti2Precision
      3.5-4 HeLP_Vertices
    3.6 [33X[0;0YChecking solutions, calculating and checking solutions[133X
      3.6-1 HeLP_VerifySolution
      3.6-2 HeLP_FindAndVerifySolution
      3.6-3 HeLP_PossiblePartialAugmentationsOfPowers
      3.6-4 HeLP_WriteTrivialSolution
    3.7 [33X[0;0YThe Wagner test[133X
      3.7-1 HeLP_WagnerTest
    3.8 [33X[0;0YAction of the automorphism group[133X
      3.8-1 HeLP_AutomorphismOrbits
    3.9 [33X[0;0YOutput[133X
      3.9-1 HeLP_PrintSolution
    3.10 [33X[0;0YEigenvalue multiplicities and character values[133X
      3.10-1 HeLP_MultiplicitiesOfEigenvalues
      3.10-2 HeLP_CharacterValue
    3.11 [33X[0;0YCheck whether Zassenhaus Conjecture is known from theoretical results[133X
      3.11-1 HeLP_IsZCKnown
  4 [33X[0;0YExtended examples[133X
    4.1 [33X[0;0YThe Character Table Library[133X
    4.2 [33X[0;0YThe behavior of the variable HeLP sol[133X
    4.3 [33X[0;0YSaving time[133X
    4.4 [33X[0;0YUsing InfoLevels[133X
    4.5 [33X[0;0YNon-standard characters[133X
    4.6 [33X[0;0YA complete example: (PQ) for the MacLaughlin simple group[133X
  5 [33X[0;0YBackground[133X
    5.1 [33X[0;0YThe Zassenhaus Conjecture and the Prime Graph Question[133X
    5.2 [33X[0;0YPartial augmentations and the structure of HeLP sol[133X
    5.3 [33X[0;0YThe HeLP equations[133X
    5.4 [33X[0;0YThe Wagner test[133X
    5.5 [33X[0;0Ys-constant characters[133X
    5.6 [33X[0;0YKnown results about the Zassenhaus Conjecture and the Prime Graph
    Question[133X
  6 [33X[0;0YRemarks on technical problems and the implementation[133X
    6.1 [33X[0;0YMaking the HeLP-package run[133X
    6.2 [33X[0;0YHow much 4ti2 and normaliz is really there?[133X
  
  
  [32X
