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Invariante Typen in torsionsfreien, auflösbaren Gruppen endlichen Ranges. (Invariant types in torsion free soluble groups of finite rank). (German) Zbl 0681.20025
There are some well-known invariant types for torsion-free abelian groups of finite rank, the inner, outer, sum and Richman type. In the classes of R-groups, of torsion-free locally nilpotent groups, of polyrational groups and especially torsion-free nilpotent groups of finite Prüfer rank a lot of similar results can be obtained. There is e.g. an inner type in the latter class, i.e. if G is the isolated hull of the elements $$x_ 1,...,x_ n$$, then the intersection $$\cap^{n}_{i=1}t(x_ i)$$ of the types of the elements $$x_ i$$ is an invariant of the group G.
Let G be a polyrational group, i.e. $$1=G_ 0\subset G_ 1\subset...\subset G_ n=G$$ with rational quotients $$G_{i+1}/G_ i{\tilde \subset}{\mathbb{Q}}$$. Then the sum type $$ST(G)=\sum^{n- 1}_{i=0}t(G_{i+1}/G_ i)$$ is an invariant of the group G. Moreover we have e.g. a dimension formula $$ST(AB)+ST(A\cap B)=ST(A)+ST(B)$$ if A and B are normal subgroups with isolated intersection.
Reviewer: O.Mutzbauer

##### MSC:
 20F16 Solvable groups, supersolvable groups 20F18 Nilpotent groups 20F19 Generalizations of solvable and nilpotent groups 20F12 Commutator calculus
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##### References:
  A. R. Asasyan, SolvableR-Groups of Finite Rank. Moscow Univ. Math. Bull. (2)37, 93-98 (1982). · Zbl 0521.20020  G. Baumslag, Some Aspects of Groups with Unique Roots. Acta Math.104, 217-303 (1960). · Zbl 0178.34901  L.Fuchs, Infinite Abelian Groups I +=II. New York 1970, 1973. · Zbl 0209.05503  P. Hall, A Contribution to the Theory of Groups of Prime-Power Order. Proc. London Math. Soc. (2)36, 29-95 (1934). · Zbl 0007.29102  A. G.Kurosh, The Theory of Groups, 2nd volume, 2nd edition. New York 1960. · Zbl 0094.24501  D. Meier andA. Rhemtulla, Rank Restricting Properties of Finitely Generated Soluble Groups. Arch. Math.44, 216-224 (1985). · Zbl 0547.20030  O. Mutzbauer, Type Invariants of Torsion-Free Abelian Groups. In: Abelian Group Theory, Proceedings, Perth 1989, Contemporary Math.87, 133-154 (1989). · Zbl 0677.20042  A. H. Rhemtulla andB. A. F. Wehrfritz, Isolators in Soluble Groups of Finite Rank. Rocky Mountain J. Math.14, 415-421 (1984). · Zbl 0545.20026  D. J. S.Robinson, Finiteness Conditions and Generalized Soluble Groups. 2 B?nde, Berlin-Heidelberg-New York 1972. · Zbl 0243.20033  D. J. S.Robinson, A Course in the Theory of Groups. Berlin-Heidelberg-New York 1982. · Zbl 0483.20001  R. B.Warfield, Jr., Nilpotent Groups. LNM513, Berlin-Heidelberg-New York 1976.  D. I. Zai?ev, On Soluble Groups of Finite Rank. Soviet Math. Dokl.9, 783-785 (1968).
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