# Finitarily hypernormalized subgroup

From Groupprops

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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]

## Definition

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

- .
- .

Then, is -hypernormalized if for some natural number . If there exists a natural number such that is -hypernormalized, we say that is finitarily hypernormalized.

## Relation with other properties

### Stronger properties

### Weaker properties

- Subnormal subgroup: In fact, a -hypernormalized subgroup is -subnormal. However, a -subnormal subgroup need not be finitarily hypernormalized, and even if it is, it need not be -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 -subnormal subgroup need not be -hypernormalized.

## Metaproperties

### Transitivity

NO:This subgroup property isnottransitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole groupABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitiveABOUT 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 doesnotsatisfy 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.ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup conditionABOUT INTERMEDIATE SUBGROUP CONDITION: View other subgroup properties not satisfying intermediate subgroup condition| View facts about intermediate subgroup condition