HN-group
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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This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]
The version of this for finite groups is at: finite HN-group
History
Origin
The term HN-group or hypernormalizing group was introduced by Alan R. Camina as part of his Ph.D. thesis in 1967. His first paper on the subject appeared in Math, Z. 100, Pages 59 - 68.
Definition
Symbol-free definition
A group is termed an HN-group or hypernormalizing group if the hypernormalizer of any ascendant subgroup is the whole group, or equivalently if every ascendant subgroup is hypernormalized.
For a finite group, this is equivalent to demanding that the hypernormalizer of any subnormal subgroup be the whole group, or equivalently, that any subnormal subgroup is finitarily hypernormalized.
Definition with symbols
A group is termed an HN-group or hypernormalizing group if for any ascendant subgroup , the hypernormalizer of , viz the limit of the normalizer sequence for in , is the whole of .
A finite group is termed an HN-group if for any subnormal subgroup , the sequence where and reaches in finitely many steps.
Formalisms
In terms of the subgroup property collapse operator
This group property can be defined in terms of the collapse of two subgroup properties. In other words, a group satisfies this group property if and only if every subgroup of it satisfying the first property (ascendant subgroup) satisfies the second property (hypernormalized subgroup), and vice versa.
View other group properties obtained in this way
We can define a HN-group as a group for which ascendant subgroup = hypernormalized subgroup
For finite groups, this is the same as saying subnormal subgroup = hypernormalized subgroup
Relation with other properties
Stronger properties
- Abelian group: Here, every subgroup is normal
- Dedekind group: Here, every subgroup is normal
- T-group (when we are working with finite groups): Here, every subnormal subgroup is normal
Conjunction with other properties
- Nilpotent HN-group: Conjunction of being nilpotent and a HN-group. For finite groups, this has the proeprty that the hypernormalizer of any subgroup is the whole group
- Solvable HN-group
References
- Hyper-normalizing groups by Alan R. Camina, Math, Z. 100, Pages 59 - 68.
- Finite soluble Hypernormalizing groups by Alan R. Camina, Journal of Algebra, 8, 362-375 (1968)
- Hypernormalizing groups by Hermann Heineken, received in 1987