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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Turing degrees of nonabelian groups


Authors: M. A. Dabkowska, M. K. Dabkowski, V. S. Harizanov and A. S. Sikora
Journal: Proc. Amer. Math. Soc. 135 (2007), 3383-3391
MSC (2000): Primary 03C57, 03D45
Published electronically: May 14, 2007
MathSciNet review: 2322771
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Abstract: For a countable structure $ \mathcal{A}$, the (Turing) degree spectrum of $ \mathcal{A}$ is the set of all Turing degrees of its isomorphic copies. If the degree spectrum of $ \mathcal{A}$ has the least degree $ \mathbf{d}$, then we say that $ \mathbf{d}$ is the (Turing) degree of the isomorphism type of $ \mathcal{A}$. So far, degrees of the isomorphism types have been studied for abelian and metabelian groups. Here, we focus on highly nonabelian groups. We show that there are various centerless groups whose isomorphism types have arbitrary Turing degrees. We also show that there are various centerless groups whose isomorphism types do not have Turing degrees.


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Additional Information

M. A. Dabkowska
Affiliation: Department of Mathematics, George Washington University, Washington, D.C. 20052
Email: gdab@gwu.edu

M. K. Dabkowski
Affiliation: Department of Mathematical Sciences, University of Texas at Dallas, Richardson, Texas 75083
Email: mdab@utdallas.edu

V. S. Harizanov
Affiliation: Department of Mathematics, George Washington University, Washington, D.C. 20052
Email: harizanv@gwu.edu

A. S. Sikora
Affiliation: Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14260
Email: asikora@buffalo.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08845-4
PII: S 0002-9939(07)08845-4
Keywords: Computable structure, Turing degree, degree spectrum, isomorphism, group
Received by editor(s): March 21, 2006
Received by editor(s) in revised form: July 1, 2006
Published electronically: May 14, 2007
Communicated by: Julia Knight
Article copyright: © Copyright 2007 American Mathematical Society