Group satisfying normalizer condition
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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition
A group is said to satisfy the normalizer condition, if it satisfies the following equivalent conditions:
- The normalizer of any proper subgroup properly contains it
- There is no proper self-normalizing subgroup of
- Every subgroup of is ascendant
Groups satisfying the normalizer condition have been termed N-groups but the term N-group is also used for groups with a particular condition on normalizers of solvable subgroups.
Metaproperties
| Metaproperty name | Satisfied? | Proof | Statement with symbols |
|---|---|---|---|
| subgroup-closed group property | Yes | If is a group satisfying normalizer condition, and is a subgroup of , then also satisfies normalizer condition. | |
| quotient-closed group property | Yes | If is a group satisfying normalizer condition, and is a normal subgroup of , then the quotient group also satisfies normalizer condition. |
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| nilpotent group | nilpotent implies normalizer condition | normalizer condition not implies nilpotent | |FULL LIST, MORE INFO | |
| group in which every subgroup is subnormal | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Gruenberg group | every cyclic subgroup is ascendant | |FULL LIST, MORE INFO | ||
| locally nilpotent group | every finitely generated subgroup is nilpotent | notmalizer condition implies locally nilpotent | locally nilpotent not implies normalizer condition | |FULL LIST, MORE INFO |
| group having no proper abnormal subgroup | there is no proper abnormal subgroup | |FULL LIST, MORE INFO | ||
| group in which every maximal subgroup is normal | every maximal subgroup is a normal subgroup | |FULL LIST, MORE INFO |
Metaproperties
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