HN-group

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.

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) reaches $$G$$ in finitely many steps.

Formalisms
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

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