Subgroup whose normal closure is fully invariant
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
- The trivial subgroup satisfies this property in every group.
- Every group satisfies this property in itself.
Important classes of examples
- All subgroups inside simple groups, as well as all contranormal subgroups, satisfy this property.
- Sylow subgroups, as well as subgroups arising as a join of Sylow subgroups (and in particular Hall subgroups) satisfy this property.
Relation with other properties
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|closure-characteristic subgroup||normal closure is a characteristic subgroup||fully invariant implies characteristic||follows by combining characteristic not implies fully invariant and the fact that, since characteristic implies normal, a characteristic subgroup is its own normal closure|||FULL LIST, MORE INFO|
|subgroup whose characteristic closure is fully invariant||characteristic closure is a fully invariant subgroup||if the normal closure is fully invariant, it is also characteristic (because fully invariant implies characteristic), and since characteristic implies normal, it is the characteristic closure||proper nontrivial subgroup in a characteristically simple group, e.g., Z2 in V4|||FULL LIST, MORE INFO|