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Type graph. (English) Zbl 0542.20032

Abelian group theory, Proc. Conf., Honolulu 1983, Lect. Notes Math. 1006, 228-252 (1983).
[For the entire collection see Zbl 0509.00009.]
This paper is a significant contribution to the theory of finite rank torsion-free abelian groups. The author defines two quasi-isomorphism invariants of such a group. The first is the sum of a type series, which is a sequence of types of factors of a composition series (a maximal length chain of pure subgroups). The sum in question is the usual sum of types, except that infinities are counted with their multiplicity. One application of this invariant is a description of groups with a unique type series.
The author’s second invariant of a group G is the type graph, which is a poset whose elements are equivalence classes of pure subgroups, where A and B are equivalent if they have the same sets of type series and so do the factors G/A and G/B. The order is defined by continuation of type series. The applications are to p-local, quotient-divisible, Butler and completely anisotropic groups and to groups all of whose torsion-free factors are homogeneous.
Reviewer: Ph.Schultz

MSC:

20K15 Torsion-free groups, finite rank
20K27 Subgroups of abelian groups

Citations:

Zbl 0509.00009