Descendant subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed descendant if we have subgroups $$H_\alpha$$ of $$G$$ for every ordinal $$\alpha$$ such that:


 * $$H_0 = G$$
 * $$H_{\alpha + 1} \ \underline{\triangleleft} \ H_\alpha$$ (viz $$H_{\alpha + 1}$$ is a normal subgroup of $$H_\alpha$$) for every ordinal $$\alpha$$.
 * If $$\alpha$$ is a limit ordinal, then $$H_\alpha = \bigcap_{\gamma < \alpha} H_\gamma$$.

and such that there is some ordinal $$\beta$$ such that $$H_\beta = H$$.

In terms of the descendant closure operator
The subgroup property of being an descendant subgroup is obtained by applying the descendant closure operator to the subgroup property of being normal.

Stronger properties

 * Normal subgroup
 * Subnormal subgroup
 * Hypernormalized subgroup

Weaker properties

 * Serial subgroup

Related properties

 * Ascendant subgroup

Opposites

 * Contranormal subgroup

Descendant-contranormal factorization
This result states that given any subgroup $$H$$ of $$G$$, there is a unique subgroup $$K$$ containing $$H$$ such that $$H$$ is contranormal in $$K$$ and $$K$$ is descendant in $$G$$.