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

Andreas Bächle () and Leo Margolis ()
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Andreas Bächle: Vrije Universiteit Brussel
Leo Margolis: Vrije Universiteit Brussel

Indian Journal of Pure and Applied Mathematics, 2021, vol. 52, issue 3, 669-686

Abstract: 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.

Keywords: Integral group ring; Unit group; Zassenhaus conjecture; Prime Graph Question; 16U60; 20C05; 20C15; 20C20 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s13226-021-00180-y

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