Partial augmentations and Brauer character values of torsion units in group rings

2006/12/15 by Martin Hertweck, Hertweck, Martin · 9 citations
Computer Science · Mathematics · #16S34 #16U60 (Primary) 20C05 (Secondary) #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.RA #math.RT #msc:16S34 #msc:16U60 #msc:20C05

paper · pdf · doi:10.48550/arxiv.math/0612429

16 pages, LateX. Thoroughly revised. Added references. Changed presentation of modular method. Lemma 5.6 replaced by Proposition 2.2

openalex publication_date 2006/12/15 · arxiv created 2007/03/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a torsion unit u of the integral group ring ℤ G of a finite group G, and a prime p which does not divide the order of u (but the order of G), a relation between the partial augmentations of u on the p-regular classes of G and Brauer character values is noted, analogous to the obvious relation between partial augmentations and ordinary character values. For non-solvable G, consequences concerning rational conjugacy of u to a group element are discussed, considering as examples the symmetric group S₅ and the groups \rm PSL(2,p^f).

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• The N-Prime Graph Question is equivalent to the Prime Graph Question
• Zassenhaus Conjecture on torsion units holds for PSL(2,p) with p a Fermat or Mersenne prime
• Integral group ring of the McLaughlin simple group

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