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 G is said to satisfy the normalizer condition, if it satisfies the following equivalent conditions:

  1. The normalizer N_G(H) of any proper subgroup H properly contains it
  2. There is no proper self-normalizing subgroup of G
  3. Every subgroup of G 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 G is a group satisfying normalizer condition, and H is a subgroup of G, then H also satisfies normalizer condition.
quotient-closed group property Yes If G is a group satisfying normalizer condition, and H is a normal subgroup of G, then the quotient group G/H 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

Metaproperties

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