Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

On free subgroups in division rings


Authors: Jason P. Bell and Jairo Gonçalves
Journal: Proc. Amer. Math. Soc. 148 (2020), 1953-1962
MSC (2010): Primary 12E15, 16K40, 20E05
DOI: https://doi.org/10.1090/proc/14888
Published electronically: January 29, 2020
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ K$ be a field, let $ \sigma $ be an automorphism, and let $ \delta $ be a $ \sigma $-derivation of $ K$. We show that the multiplicative group of nonzero elements of the division ring $ D=K(x;\sigma ,\delta )$ contains a free noncyclic subgroup unless $ D$ is commutative, answering a special case of a conjecture of Lichtman. As an application, we show that division algebras formed by taking the Goldie ring of quotients of group algebras of torsion-free nonabelian solvable-by-finite groups always contain free noncyclic subgroups.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 12E15, 16K40, 20E05

Retrieve articles in all journals with MSC (2010): 12E15, 16K40, 20E05


Additional Information

Jason P. Bell
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Email: jpbell@uwaterloo.ca

Jairo Gonçalves
Affiliation: Department of Mathematics, University of Saõ Paulo, Saõ Paulo, SP, 05508-090, Brazil
Email: jz.goncalves@usp.br

DOI: https://doi.org/10.1090/proc/14888
Keywords: Division rings, free groups, group algebras, solvable-by-finite groups, Ore extensions
Received by editor(s): December 13, 2018
Received by editor(s) in revised form: September 30, 2019
Published electronically: January 29, 2020
Communicated by: Jerzy Weyman
Article copyright: © Copyright 2020 American Mathematical Society