Groups with \(H/\text{core}(H)\) satisfying max or min for all subgroups \(H\). (English) Zbl 1073.20032
Summary: Let \(G\) be a group in which \(H/H_G\) satisfies min, the minimal condition on subgroups, for all subgroups \(H\) of \(G\), where \(H_G\) denotes the normal core of \(H\) in \(G\). We show that, with the additional hypothesis that \(G\) has all of its periodic images locally finite, \(G\) has an Abelian normal subgroup \(A\) such that \(G/A\) has min; further consequences are then established. With the maximal condition replacing the minimal condition, a similar conclusion does not hold: we give an example of a (torsion-free) nilpotent group \(G\) such that \(H/H_G\) satisfies max for all subgroups \(H\) but \(G\) is not Abelian-by-max.
MSC:
| 20F22 | Other classes of groups defined by subgroup chains |
| 20E07 | Subgroup theorems; subgroup growth |
| 20F18 | Nilpotent groups |
Keywords:
minimal condition on subgroups; locally finite periodic images; Abelian normal subgroups; maximal condition; torsion-free nilpotent groupsReferences:
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