Dualities for self–small groups

Breaz, Simion; Schultz, Phill

doi:10.1090/S0002-9939-2011-10919-5

Dualities for self–small groups
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by Simion Breaz and Phill Schultz PDF

Proc. Amer. Math. Soc. 140 (2012), 69-82 Request permission

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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