Hereditarily characteristic subgroup
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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]
This article describes a property that arises as the conjunction of a subgroup property: normal-hereditarily characteristic subgroup with a group property (itself viewed as a subgroup property): Dedekind group
View a complete list of such conjunctions
Definition
A subgroup of a group is termed a hereditarily characteristic subgroup if every subgroup of it is a characteristic subgroup in the whole group.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Cyclic characteristic subgroup | cyclic group and characteristic subgroup of whole group | cyclic characteristic implies hereditarily characteristic | hereditarily characteristic not implies cyclic in finite |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Normal-hereditarily characteristic subgroup | every normal subgroup of the subgroup is characteristic in the whole group | |||
| Hereditarily normal subgroup | every subgroup is normal in the whole group |
Metaproperties
Left realization
For a group to be realizable as a hereditarily characteristic subgroup in a bigger group , a necessary condition is that be a Dedekind group, i.e., every subgroup of must be normal in .
