HN-group

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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 $G$ is termed an HN-group or hypernormalizing group if for any ascendant subgroup $H$, the hypernormalizer of $H$, viz the limit of the normalizer sequence for $H$ in $G$, is the whole of $G$.

A finite group $G$ is termed an HN-group if for any subnormal subgroup $H$, the sequence $H_i$ where $H_0=H$ and H_{i+1} = N_G(H_i)</math> reaches $G$ in finitely many steps.

Relation with other properties

Stronger properties

• Abelian group: Here, every subgroup is normal
• Hamiltonian group: Here, every subgroup is normal
• T-group: Here, every subnormal subgroup is normal

Conjunctions 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