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Groups with many nilpotent subgroups. (English) Zbl 0861.20039

The paper is in the spirit of a sequence of papers that deal with properties of groups in which every infinite subset of elements contains a finite subset, or a finite subset with \(k\) elements, that generates a subgroup of some prescribed kind. The authors consider groups in which every infinite set of elements contains \(k-1\) elements that generate a nilpotent subgroup of class at most \(k\), which they call \((\text{N}_k)\) groups, and more generally groups in which every infinite set of elements contains a finite subset \(X\) that generates a nilpotent subgroup of class less than the cardinality of \(X\), which they call (N) groups. Their main theorems [slightly reworded] are Theorem A: A finitely generated group \(G\) is an \((\text{N}_k)\) group if, and only if, it is finite-by-nilpotent of class at most \(k\), and if and only if the \(k\)-th term of the upper central series \(Z_k(G)\) has finite index in \(G\). Theorem B: A torsion-free \((\text{N}_k)\) group is nilpotent of class at most \(k\) and a torsion free (N) group is hypercentral: on the other hand a torsion (N) group is hypercentral-by-finite. Theorem C: An \((\text{N}_k)\) group is an FC-group if, and only if, it contains a subgroup of finite index every pair of elements of which generates a nilpotent subgroup of class at most \(k-1\): moreover, in an FC-group which is an (N) group the \(\omega\)th term of the upper central series has finite index. There are also some generalisations of these results, and a discussion of the role of Ramsey’s Theorem often used in the earlier papers. However, the underlying set theory necessarily contains the Axiom of Choice.

MSC:

20F19 Generalizations of solvable and nilpotent groups
20F18 Nilpotent groups
20F24 FC-groups and their generalizations
20F05 Generators, relations, and presentations of groups
20E07 Subgroup theorems; subgroup growth
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References:

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