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From examples to methods: two cases from the study of units in integral group rings

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Abstract

In this article, we review the proofs of the first Zassenhaus Conjecture on conjugacy of torsion units in integral group rings for the alternating groups of degree 5 and 6, by Luthar-Passi and Hertweck. We describe how the study of these examples led to the development of two methods – the HeLP method and the lattice method. We exhibit these methods and summarize some results which were achieved using them. We then apply these methods to the study of the first Zassenhaus conjecture for the alternating group of degree 7 where only one critical case remains open for a full answer. Along the way we show in examples how recently obtained results can be combined with the methods presented and collect open problems some of which could be attacked using these methods.

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Notes

  1. This package is free and open software, comes with recent GAP installations and is also available from https://gap-packages.github.io/HeLP/. Note that some of the external software might need compilation, see the manual.

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Authors and Affiliations

  1. Vakgroep Wiskunde, Vrije Universiteit Brussel, Pleinlaan 2, 1050, Brussels, Belgium

    Andreas Bächle & Leo Margolis

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  1. Andreas Bächle
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  2. Leo Margolis
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Correspondence to Andreas Bächle.

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Communicated by Gadadhar Misra.

The second author is a postdoctoral researcher of the Research Foundation Flanders (FWO-Vlaanderen).

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Bächle, A., Margolis, L. From examples to methods: two cases from the study of units in integral group rings. Indian J Pure Appl Math 52, 669–686 (2021). https://doi.org/10.1007/s13226-021-00180-y

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