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Finite rank Butler groups: A survey of recent results. (English) Zbl 0804.20043
Fuchs, Laszlo (ed.) et al., Abelian groups. Proceedings of the 1991 Curaçao conference. New York: Marcel Dekker, Inc. Lect. Notes Pure Appl. Math. 146, 17-41 (1993).
This paper is among other things an annotated guide to literature with 51 references, all written in the last twenty years except for a few classics, many not yet in print. It is more helpful for people in the know who want to be updated than for the novice who wants to be introduced to the subject. Sixteen unsolved problems are suggested for study. Almost half of the paper is devoted to the groups \(G[A]\) which are essentially quotients of (finite rank) completely decomposable groups modulo a pure rank-one subgroup. This is a special class but there are many results obtained by different people with different approaches. It is important as a source of examples and insights. While quasi- isomorphism dominates the sections on the groups \(G[A]\), another section deals with near-isomorphism and representations of (again rather special) Butler groups. One is lead to representations over quotient rings of \(\mathbb{Z}\). Since there is little or no literature about modules over rings \(\mathbb{Z}/n\mathbb{Z}\) with distinguished submodules, there is a better chance for abelian group theory to promote such modules than the other way around. The final six and a half pages are devoted to almost completely decomposable groups. The topic is approached via concrete examples and is a nice source of such. The main concern are recent classification results up to near-isomorphism.
For the entire collection see [Zbl 0778.00023].
Reviewer: A.Mader (Honolulu)

MSC:
20K15 Torsion-free groups, finite rank
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
20K25 Direct sums, direct products, etc. for abelian groups
20-02 Research exposition (monographs, survey articles) pertaining to group theory
20K27 Subgroups of abelian groups
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