Hypernormalized subgroup
From Groupprops
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof.
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This is a variation of normality
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Contents |
Definition
Symbol-free definition
A subgroup of a group is said to be hypernormalized if its hypernormalizer is the whole group. Here, the hypernormalizer is the limit of the transfinite ascending chain that begins with the subgroup and where each successor is the normalizer of its predecessor in the whole group.
If the hypernormalizer sequence reachse the whole group in finitely many steps, we call the subgroup finitarily hypernormalized.
Definition with symbols
A subgroup H of a group G is said to be hypernormalized if the sequence Hα ends at G, where <math.H_\alpha</math> is defined as follows:
- H0 = H
- Hα + 1 = NG(Hα)
- Hα is the ascending chain union of Hβ for β < α when α is a limit ordinal
If Hr = G for a finite integer r, we say that H is finiarily hypernormalized in G.
Relation with other properties
Stronger properties
Weaker properties
Related group properties
- HN-group is a group where every subgroup is 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
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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
A hypernormalized subgroup of a hypernormalized subgroup need not be hypernormalized. This can be seen from the fact that there are subnormal subgroups which are not hypernormalized.
