Hypernormalized subgroup

Symbol-free definition
A subgroup of a group is said to be hypernormalized if its hypernormalizer is the whole group. Here, the hypernormalizer is the limit of the transfinite ascending chain that begins with the subgroup and where each successor is the normalizer of its predecessor in the whole group.

If the hypernormalizer sequence reachse the whole group in finitely many steps, we call the subgroup finitarily hypernormalized.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is said to be hypernormalized if the sequence $$H_{\alpha}$$ ends at $$G$$, where <math.H_\alpha is defined as follows:


 * $$H_0 = H$$
 * $$H_{\alpha + 1} = N_G(H_{\alpha})$$
 * $$H_\alpha$$ is the ascending chain union of $$H_\beta$$ for $$\beta < \alpha$$ when $$\alpha$$ is a limit ordinal

If $$H_r = G$$ for a finite integer $$r$$, we say that $$H$$ is finiarily hypernormalized in $$G$$.

Stronger properties

 * Weaker than::Finitarily hypernormalized subgroup
 * Weaker than::Normal subgroup
 * Weaker than::2-hypernormalized subgroup

Weaker properties

 * Stronger than::Ascendant subgroup
 * Stronger than::Serial subgroup

Related group properties

 * HN-group is a group where every subgroup is hypernormalized.

Metaproperties
A hypernormalized subgroup of a hypernormalized subgroup need not be hypernormalized. This can be seen from the fact that there are subnormal subgroups which are not hypernormalized.