On free subgroups in division rings
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- by Jason P. Bell and Jairo Gonçalves PDF
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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
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Additional Information
- Jason P. Bell
- Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
- MR Author ID: 632303
- Email: jpbell@uwaterloo.ca
- Jairo Gonçalves
- Affiliation: Department of Mathematics, University of Saõ Paulo, Saõ Paulo, SP, 05508-090, Brazil
- MR Author ID: 75040
- Email: jz.goncalves@usp.br
- 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
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 1953-1962
- MSC (2010): Primary 12E15, 16K40, 20E05
- DOI: https://doi.org/10.1090/proc/14888
- MathSciNet review: 4078080