Group satisfying normalizer condition

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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

• Nilpotent group: It turns out that for a finitely generated group, the two properties are equivalent. For proof of the implication, refer Nilpotent implies normalizer condition and for proof of its strictness (i.e. the reverse implication being false) refer Normalizer condition not implies nilpotent.
• Group in which every subgroup is subnormal

Weaker properties

• Gruenberg group
• Locally nilpotent group
• Group having no proper abnormal subgroup
• Group in which every maximal subgroup is normal

Metaproperties

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Definition links

• Planetmath page