Ascendant subgroup

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


 * $$H_0 = H$$
 * $$H_\alpha \ \underline{\triangleleft} \ H_{\alpha + 1}$$ (viz $$H_\alpha$$ is a normal subgroup of $$H_{\alpha + 1}$$) for every ordinal $$\alpha$$.
 * If $$\alpha$$ is a limit ordinal, then $$H_\alpha = \bigcup_{\gamma < \alpha} H_\gamma$$, i.e., it is the union of all preceding subgroups. Note that the union of any ascending chain of subgroups is indeed a sugbroup (in fact, more generally, directed union of subgroups is subgroup). We can also define $$H_{\alpha}$$as $$\langle H_\gamma \rangle_{\gamma < \alpha}$$, i.e., it is the join of all preceding subgroups.

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

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

Opposites

 * Self-normalizing subgroup