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