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Dualities for self-small groups

Authors: Simion Breaz and Phill Schultz
Journal: Proc. Amer. Math. Soc. 140 (2012), 69-82
MSC (2010): Primary 20K21, 20K30, 20K40
Published electronically: May 12, 2011
MathSciNet review: 2833518
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Abstract: We construct a family of dualities on some subcategories of the quasi-category $ \mathcal{S}$ of self-small groups of finite torsion-free rank which cover the class $ \mathcal{S}$. These dualities extend several of those in the literature. As an application, we show that a group $ A\in\mathcal{S}$ is determined up to quasi-isomorphism by the $ \mathbb{Q}$-algebras $ \{\mathbb{Q}\operatorname{Hom}(C,A): C\in\mathcal{S}\}$ and $ \{\mathbb{Q}\operatorname{Hom}(A,C): C\in\mathcal{S}\}$. We also generalize Butler's Theorem to self-small mixed groups of finite torsion-free rank.

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Additional Information

Simion Breaz
Affiliation: Faculty of Mathematics and Computer Science, “Babeş-Bolyai” University, Str. Mihail Kogălniceanu 1, 400084 Cluj-Napoca, Romania

Phill Schultz
Affiliation: School of Mathematics and Statistics, The University of Western Australia, Nedlands, 6009, Australia

Keywords: Self-small abelian group, finite rank torsion-free group, quotient divisible group, quasi-homomorphism category
Received by editor(s): March 31, 2010
Received by editor(s) in revised form: November 8, 2010
Published electronically: May 12, 2011
Additional Notes: The first author is supported by the UEFISCSU-CNCSIS, grant ID489
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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