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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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
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
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 with symbols
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
Relation with other 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
- 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