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Involutions and free pairs of bicyclic units in integral group rings of non-nilpotent groups


Authors: J. Z. Gonçalves and D. S. Passman
Journal: Proc. Amer. Math. Soc. 143 (2015), 2395-2401
MSC (2010): Primary 16S34, 20D15, 20E05
DOI: https://doi.org/10.1090/S0002-9939-2015-12550-6
Published electronically: February 3, 2015
MathSciNet review: 3326022
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Abstract: If $ {}^*\colon G\to G$ is an involution on the finite group $ G$, then $ {}^*$ extends to an involution on the integral group ring $ \mathbb{Z}[G]$. In this paper, we consider whether bicyclic units $ u\in \mathbb{Z}[G]$ exist with the property that the group $ \langle u,u^*\rangle $, generated by $ u$ and $ u^*$, is free on the two generators. If this occurs, we say that $ (u,u^*)$ is a free bicyclic pair. It turns out that the existence of $ u$ depends strongly upon the structure of $ G$ and on the nature of the involution. The main result here is that if $ G$ is a non-nilpotent group, then for any involution, $ \mathbb{Z}[G]$ contains a free bicyclic pair.


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Additional Information

J. Z. Gonçalves
Affiliation: Department of Mathematics, University of São Paulo, São Paulo, 05389-970, Brazil
Email: jz.goncalves@usp.br

D. S. Passman
Affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706
Email: passman@math.wisc.edu

DOI: https://doi.org/10.1090/S0002-9939-2015-12550-6
Keywords: Integral group ring, non-nilpotent group, dihedral group, bicyclic unit, free bicyclic pair
Received by editor(s): January 26, 2014
Published electronically: February 3, 2015
Additional Notes: This research was supported in part by the grant CNPq 303.756/82-5 and by Fapesp-Brazil, Proj. Tematico 00/07.291-0
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2015 American Mathematical Society