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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Free products in the unit group of the integral group ring of a finite group
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by Geoffrey Janssens, Eric Jespers and Doryan Temmerman PDF
Proc. Amer. Math. Soc. 145 (2017), 2771-2783 Request permission

Abstract:

Let $G$ be a finite group and let $p$ be a prime. We continue the search for generic constructions of free products and free monoids in the unit group $\mathcal {U}(\mathbb {Z}G)$ of the integral group ring $\mathbb {Z} G$. For a nilpotent group $G$ with a non-central element $g$ of order $p$, explicit generic constructions are given of two periodic units $b_1$ and $b_2$ in $\mathcal {U}(\mathbb {Z}G)$ such that $\langle b_1, b_2\rangle =\langle b_1\rangle \star \langle b_2 \rangle \cong \mathbb {Z}_p \star \mathbb {Z}_{p}$, a free product of two cyclic groups of prime order. Moreover, if $G$ is nilpotent of class $2$ and $g$ has order $p^n$, then also concrete generators for free products $\mathbb {Z}_{p^k} \star \mathbb {Z}_{p^m}$ are constructed (with $1\leq k,m\leq n$). As an application, for finite nilpotent groups, we obtain earlier results of Marciniak-Sehgal and Gonçalves-Passman. Further, for an arbitrary finite group $G$ we give generic constructions of free monoids in $\mathcal {U}(\mathbb {Z}G)$ that generate an infinite solvable subgroup.
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Additional Information
  • Geoffrey Janssens
  • Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan $2$, B-1050 Elsene, Belgium
  • MR Author ID: 1005538
  • Email: Geoffrey.Janssens@vub.be
  • Eric Jespers
  • Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan $2$, B-1050 Elsene, Belgium
  • MR Author ID: 94560
  • Email: Eric.Jespers@vub.be
  • Doryan Temmerman
  • Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan $2$, B-1050 Elsene, Belgium
  • Email: Doryan.Temmerman@vub.be
  • Received by editor(s): July 8, 2016
  • Published electronically: April 4, 2017
  • Additional Notes: The first and third authors were supported by Fonds voor Wetenschappelijk Onderzoek (Flanders)
    The second author was supported by Onderzoeksraad of Vrije Universiteit Brussel and Fonds voor Wetenschappelijk Onderzoek (Flanders)
  • Communicated by: Pham Huu Tiep
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 2771-2783
  • MSC (2010): Primary 16U60, 20C05, 16S34, 20E06; Secondary 20C10, 20C40
  • DOI: https://doi.org/10.1090/proc/13631
  • MathSciNet review: 3637929