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 of finite rank arises as the middle group of an extension where is free of finite rank and is torsion with the -ranks of finite for all primes . Given such a and we study the equivalence classes of such extensions which result from stipulating that two extensions , are equivalent if for and . We reduce the problem to -primary of finite rank, where in the one case is injective, and in the other case is reduced. Suppose . 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 -generated subgroups of where . Two -generated subgroups of will be called equivalent if one can be mapped onto the other by an automorphism of .
-  L. Fuchs, Infinite abelian groups. Vols. I, II, Pure and Appl. Math., vol. 36, Academic Press, New York, 1970, 1973. MR 41 #333.
-  S. A. Khabbaz and E. H. Toubassi, The module structure of 𝐸𝑥𝑡(𝐹,𝑇) over the endomorphism ring of 𝑇, Pacific J. Math. 54 (1974), 169–176. MR 0360759
-  S. A. Khabbaz and E. H. Toubassi, 𝐸𝑥𝑡(𝐴,𝑇) as a module over 𝐸𝑛𝑑(𝑇), Proc. Amer. Math. Soc. 48 (1975), 269–275. MR 0360865, https://doi.org/10.1090/S0002-9939-1975-0360865-4
- L. Fuchs, Infinite abelian groups. Vols. I, II, Pure and Appl. Math., vol. 36, Academic Press, New York, 1970, 1973. MR 41 #333.
- S. A. Khabbaz and E. H. Toubassi, The module structure of over the endomorphism ring of , Pacific J. Math. 54 (1975), 169-176. MR 0360759 (50:13206)
- -, as a module over , Proc. Amer. Math. Soc. 48 (1975), 269-275. MR 0360865 (50:13312)
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Keywords: Module of extensions, finite rank, homological methods in group theory
Article copyright: © Copyright 1975 American Mathematical Society