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Groups whose proper subgroups are locally finite-by-nilpotent. (English) Zbl 1131.20023

Let \(\mathcal X\) be a class of groups. If a group \(G\) does not belong to \(\mathcal X\), but every of its proper subgroups belongs to \(\mathcal X\), then \(G\) is said to be a minimal non-\(\mathcal X\)-group. One of the two main results of the article yields that if \(G\) is a minimal non-(locally finite-by-nilpotent)-group, then \(G\) is a finitely generated perfect group which has no non-trivial finite factor and its factor-group by the Frattini subgroup is an infinite simple group (Theorem 1). Theorem 2 is similar concerning minimal non-(locally finite-by-nilpotent of class at most \(c\))-groups.

MSC:

20F19 Generalizations of solvable and nilpotent groups
20E25 Local properties of groups
20E32 Simple groups
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References:

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