Group satisfying normalizer condition
From Groupprops
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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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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
This is a variation of nilpotence|Find other variations of nilpotence | Read a survey article on varying nilpotence
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 | Group in which every subgroup is subnormal|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 | Gruenberg group|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 |
