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An application of blocks to torsion units in group rings


Authors: Andreas Bächle and Leo Margolis
Journal: Proc. Amer. Math. Soc. 147 (2019), 4221-4231
MSC (2010): Primary 16U60, 20C05, 20C20
DOI: https://doi.org/10.1090/proc/14561
Published electronically: July 8, 2019
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Abstract: We use the theory of blocks of cyclic defect to prove that under a certain condition on the principal $ p$-block of a finite group $ G$, the normalized unit group of the integral group ring of $ G$ contains an element of order $ pq$ if and only if so does $ G$ for $ q$ a prime different from $ p$. Using this we verify the Prime Graph Question for all alternating and symmetric groups and also for two sporadic simple groups.


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Additional Information

Andreas Bächle
Affiliation: Vakgroep Wiskunde, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium
Email: andreas.bachle@vub.be

Leo Margolis
Affiliation: Vakgroep Wiskunde, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium
Email: leo.margolis@vub.be

DOI: https://doi.org/10.1090/proc/14561
Keywords: Integral group ring, Prime Graph Question, cyclic defect, symmetric groups
Received by editor(s): September 18, 2018
Received by editor(s) in revised form: January 8, 2019, and January 21, 2019
Published electronically: July 8, 2019
Additional Notes: Both authors are postdoctoral researchers of the Research Foundation Flanders (FWO - Vlaanderen).
Dedicated: One may dare to ask whether the theory of cyclic blocks can provide additional insight into Zassenhaus’ conjecture (ZC1). We have no clue, but...\textsuperscript{1}
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2019 American Mathematical Society