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