Finitarily hypernormalized subgroup

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

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Definition

A subgroup $H$ of a group $G$ is termed a finitarily hypernormalized subgroup if the iterated sequence of normalizers, starting from $H$, reaches $G$ in finitely many steps. In other words, define a sequence $H_i$ by:

• $H_0=H$.
• $H_{i+1}=N_G(H_i)$.

Then, $H$ is $k$-hypernormalized if $H_k=G$ for some natural number $k$. If there exists a natural number $k$ such that $H$ is $k$-hypernormalized, we say that $H$ is finitarily hypernormalized.

Relation with other properties

Stronger properties

• Normal subgroup
• 2-hypernormalized subgroup
• Intermediately finitarily hypernormalized subgroup

Weaker properties

• Subnormal subgroup: In fact, a $k$-hypernormalized subgroup is $k$-subnormal. However, a $k$-subnormal subgroup need not be finitarily hypernormalized, and even if it is, it need not be $k$-normalized. For full proof, refer: Abnormal normalizer and 2-subnormal not implies normal, normalizer of 2-subnormal subgroup may have arbitrarily large subnormal depth
• Hypernormalized subgroup

Related group properties

• Hypernormalizing group is a group in which every ascendant subgroup is hypernormalized. A finite hypernormalizing group, or more generally, a slender hypernormalizing group, must have every subnormal subgroup is finitarily hypernormalized.
• In a nilpotent group every subgroup is hypernormalized, but a $k$-subnormal subgroup need not be $k$-hypernormalized.

Metaproperties

Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
ABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

Intermediate subgroup condition

NO: This subgroup property does not satisfy the intermediate subgroup condition: it is possible to have a subgroup satisfying the property in the whole group but not satisfying the property in some intermediate subgroup.
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ABOUT INTERMEDIATE SUBGROUP CONDITION: View other subgroup properties not satisfying intermediate subgroup condition| View facts about intermediate subgroup condition