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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.
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VIEW RELATED: | Subgroup property non-implications | | | |
RANDOM TIP:The relation with other properties section lists stronger and weaker properties, along with links to proofs of the implications and non-implications. This helps give a feel of how the subgroup property relates to other properties.

This is a variation of normality
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Contents 1 Definition 1.1 Symbol-free definition 1.2 Definition with symbols 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties 2.3 Related group properties 3 Metaproperties 3.1 Transitivity

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 H_α ends at G, where <math.H_\alpha</math> is defined as follows:

• H₀ = H
• H_{α + 1} = N_G(H_α)
• H_α is the ascending chain union of H_β for β < α when α is a limit ordinal

If H_r = G for a finite integer r, we say that H is finiarily hypernormalized in G.

Relation with other properties

Stronger properties

• Finitarily hypernormalized subgroup
• Normal subgroup
• 2-hypernormalized subgroup

Weaker properties

• Ascendant subgroup
• Serial subgroup

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
View a complete list of transitive subgroup properties|View facts related to transitivity of subgroup properties

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.