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# HeLP

## Hertweck-Luthar-Passi method.

3.4

20 November 2018

Andreas Bächle
Email: ABachle@vub.ac.be
Homepage: http://homepages.vub.ac.be/~abachle/
Address:
Vrije Universiteit Brussel
Vakgroep Wiskunde
Pleinlaan 2
1050 Brussels
Belgium

Leo Margolis
Email: Leo.Margolis@vub.be
Homepage: http://homepages.vub.ac.be/~lmargoli/
Address:
Vrije Universiteit Brussel
Vakgroep Wiskunde
Pleinlaan 2
1050 Brussels
Belgium

### Copyright

© 2017 by Andreas Bächle and Leo Margolis

This package is free software and may be distributed under the terms and conditions of the GNU Public License Version 2.

### Acknowledgements

The authors are grateful to Sebastian Gutsche, Christof Söger and Max Horn for endowing GAP with a 4ti2-Interface and a normlaiz-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.

### Contents

1 Introduction
2 The main functions
2.1 Zassenhaus Conjecture
2.1-1 HeLP_ZC
2.2 Prime Graph Question
2.2-1 HeLP_PQ
3 Further functions
3.1 Checks for specific orders
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 Checks for specific orders with s-constant characters
3.2-1 HeLP_WithGivenOrderSConstant
3.2-2 HeLP_AddGaloisCharacterSums
3.3 Checks for all orders
3.3-1 HeLP_AllOrders
3.3-2 HeLP_AllOrdersPQ
3.4 Changing the used Character Table
3.4-1 HeLP_ChangeCharKeepSols
3.4-2 HeLP_Reset
3.5 Influencing how the Systems of Inequalities are solved
3.5-1 HeLP_Solver
3.5-2 HeLP_UseRedund
3.5-3 HeLP_Change4ti2Precision
3.5-4 HeLP_Vertices
3.6 Checking solutions, calculating and checking solutions
3.6-1 HeLP_VerifySolution
3.6-2 HeLP_FindAndVerifySolution
3.6-3 HeLP_PossiblePartialAugmentationsOfPowers
3.6-4 HeLP_WriteTrivialSolution
3.7 The Wagner test
3.7-1 HeLP_WagnerTest
3.8 Action of the automorphism group
3.8-1 HeLP_AutomorphismOrbits
3.9 Output
3.9-1 HeLP_PrintSolution
3.10 Eigenvalue multiplicities and character values
3.10-1 HeLP_MultiplicitiesOfEigenvalues
3.10-2 HeLP_CharacterValue
3.11 Check whether Zassenhaus Conjecture is known from theoretical results
3.11-1 HeLP_IsZCKnown
4 Extended examples
4.1 The Character Table Library
4.2 The behavior of the variable HeLP sol
4.3 Saving time
4.4 Using InfoLevels
4.5 Non-standard characters
4.6 A complete example: (PQ) for the MacLaughlin simple group
5 Background
5.1 The Zassenhaus Conjecture and the Prime Graph Question
5.2 Partial augmentations and the structure of HeLP sol
5.3 The HeLP equations
5.4 The Wagner test
5.5 s-constant characters
5.6 Known results about the Zassenhaus Conjecture and the Prime Graph Question
6 Remarks on technical problems and the implementation
6.1 Making the HeLP-package run
6.2 How much 4ti2 and normaliz is really there?
References
Index
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