Iterated agemo subgroup of p-group

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Definition

Suppose $p$ is a prime number and $G$ is a p-group. A subgroup $H$ of $G$ is an iterated agemo subgroup of $G$ if there exists a sequence $a_1,a_2,\dots ,a_r$ such that:

$H:={\mho }^{a_1}({\mho }^{a_2}({\mho }^{a_3}(\dots ({\mho }^{a_r}(G)))\dots )$.

Relation with other properties

Stronger properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
agemo subgroup of p-group			iterated agemo subgroup not implies agemo subgroup

Weaker properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
verbal subgroup of p-group		iterated agemo implies verbal	verbal not implies iterated agemo