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Groups with many hypercentral subgroups. (English) Zbl 1154.20032

The authors describe solvable groups with minimum condition for non-hypercentral (non-nilpotent) subgroups. If such a group is not Chernikov and not hypercentral (not nilpotent), then it is a direct product \(P\times Q\) of a hypercentral (nilpotent) Chernikov \(p'\)-group \(Q\) and a non-hypercentral (non-nilpotent) \(p\)-group \(P\) which contains a normal HM*-subgroup \(H\) of finite index (with \(H'\) nilpotent in the non-nilpotent case) satisfying the normalizer condition. – A group \(T\) is an HM*-group if \(T'\) is a hypercentral \(p\)-group and \(T/T'\) is a divisible Chernikov \(p\)-group.

MSC:

20F16 Solvable groups, supersolvable groups
20F19 Generalizations of solvable and nilpotent groups
20F22 Other classes of groups defined by subgroup chains
20E15 Chains and lattices of subgroups, subnormal subgroups
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