Group satisfying normalizer condition

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

Symbol-free definition

A group is termed a N-group or is said to satisfy the normalizer condition, if the normalizer of any proper subgroup properly contains it, or equivalently, if it has no proper self-normalizing subgroup.

A group is a N-group if and only if every subgroup is ascendant.

Definition with symbols

A group $G$ is termed a N-group or is said to satisfy a normalizer condition if for any proper subgroup $H$ of $G$, $H<N_G(H)$ with the inclusion being strict (that is, $H$ is properly contained in its normalizer).

Relation with other properties

Stronger properties

• Nilpotent group: It turns out that for a finitely generated group, the two properties are equivalent.

Weaker properties

• Locally nilpotent group

Metaproperties

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