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Proceedings of the American Mathematical Society

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



Extensions of abelian groups of finite rank

Authors: S. A. Khabbaz and E. H. Toubassi
Journal: Proc. Amer. Math. Soc. 50 (1975), 115-120
MSC: Primary 20K35
MathSciNet review: 0372073
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Abstract: Every abelian group $ X$ of finite rank arises as the middle group of an extension $ e:0 \to F \to X \to T \to 0$ where $ F$ is free of finite rank $ n$ and $ T$ is torsion with the $ p$-ranks of $ T$ finite for all primes $ p$. Given such a $ T$ and $ F$ we study the equivalence classes of such extensions which result from stipulating that two extensions $ {e_i}:0 \to F \to {X_i} \to T \to 0,i = 1,2$, are equivalent if $ {e_1} = \beta {e_2}\alpha $ for $ \alpha \in \operatorname{Aut} (T)$ and $ \beta \in \operatorname{Aut} (F)$. We reduce the problem to $ T$ $ p$-primary of finite rank, where in the one case $ T$ is injective, and in the other case $ T$ is reduced. Suppose $ T = \Pi _{i = 1}^m{T_i}$. In our main theorems we prove that in each case these equivalence classes of extensions are in 1-1 correspondence with the equivalence classes of $ n$-generated subgroups of $ E$ where $ E = \Pi _{i = 1}^m{E_i},{E_i} = \operatorname{End} ({T_i})$. Two $ n$-generated subgroups of $ E$ will be called equivalent if one can be mapped onto the other by an automorphism of $ E$.

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Keywords: Module of extensions, finite rank, homological methods in group theory
Article copyright: © Copyright 1975 American Mathematical Society

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