Finitarily hypernormalized subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a finitarily hypernormalized subgroup if the iterated sequence of normalizers, starting from $$H$$, reaches $$G$$ in finitely many steps. In other words, define a sequence $$H_i$$ by:


 * $$H_0 = H$$.
 * $$H_{i+1} = N_G(H_i)$$.

Then, $$H$$ is $$k$$-hypernormalized if $$H_k = G$$ for some natural number $$k$$. If there exists a natural number $$k$$ such that $$H$$ is $$k$$-hypernormalized, we say that $$H$$ is finitarily hypernormalized.

Stronger properties

 * Weaker than::Normal subgroup
 * Weaker than::2-hypernormalized subgroup
 * Weaker than::Intermediately finitarily hypernormalized subgroup

Weaker properties

 * Stronger than::Subnormal subgroup: In fact, a $$k$$-hypernormalized subgroup is $$k$$-subnormal. However, a $$k$$-subnormal subgroup need not be finitarily hypernormalized, and even if it is, it need not be $$k$$-normalized.
 * Stronger than::Hypernormalized subgroup

Related group properties

 * Hypernormalizing group is a group in which every ascendant subgroup is hypernormalized. A finite hypernormalizing group, or more generally, a slender hypernormalizing group, must have every subnormal subgroup is finitarily hypernormalized.
 * In a nilpotent group every subgroup is hypernormalized, but a $$k$$-subnormal subgroup need not be $$k$$-hypernormalized.