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Free products in the unit group of the integral group ring of a finite group


Authors: Geoffrey Janssens, Eric Jespers and Doryan Temmerman
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
Published electronically: April 4, 2017
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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
Email: Geoffrey.Janssens@vub.be

Eric Jespers
Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan $2$, B-1050 Elsene, Belgium
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

DOI: https://doi.org/10.1090/proc/13631
Keywords: Group ring, unit group, free product, generic units
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
Article copyright: © Copyright 2017 American Mathematical Society