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Locally nilpotent \(p\)-groups whose proper subgroups are hypercentral or nilpotent-by-Chernikov. (English) Zbl 0961.20031
The author’s four main results are the following. (1) Let \(G\) be a non-hypercentral Fitting \(p\)-group such that every proper subgroup of \(G\) is hypercentral and soluble. Then \(G\) is soluble. (2) Let \(G\) be a barely transitive locally nilpotent \(p\)-group. If a point stabilizer of \(G\) is hypercentral and soluble, then \(G\) is not perfect. (3) Let \(G\) be a locally nilpotent \(p\)-group with each of its proper subgroups nilpotent-by-Chernikov. Then \(G\) is nilpotent-by-Chernikov. (4) Let \(G\) be a locally nilpotent \(p\)-group with each of its proper subgroups nilpotent-by-finite. If \(G\) is not nilpotent-by-finite, then \(G\) modulo its derived subgroup is a Prüfer \(p^\infty\)-group.

MSC:
20F19 Generalizations of solvable and nilpotent groups
20F50 Periodic groups; locally finite groups
20E07 Subgroup theorems; subgroup growth
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