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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

20F16 Solvable groups, supersolvable groups
20F18 Nilpotent groups
20F19 Generalizations of solvable and nilpotent groups
20F12 Commutator calculus
Full Text: DOI
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