Completely divisibility-closed subgroup

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Definition

Suppose $G$ is a group. A subgroup $H$ of $G$ is termed completely divisibility-closed if the following holds: for any prime number $p$ such that $G$ is $p$-divisible, and any $g\in H$, all $p^{th}$ roots of $g$ in $G$ lie inside $H$.

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
transitive subgroup property	Yes	complete divisibility-closedness is transitive	If $H\leq K\leq G$ are groups such that $H$ is completely divisibility-closed in $K$ and $K$ is completely divisibility-closed in $G$, then $H$ is completely divisibility-closed in $G$.
strongly intersection-closed subgroup property	Yes	complete divisibility-closedness is strongly intersection-closed	If $H_{i},i\in I$ are completely divisibility-closed subgroups of $G$, so is $\bigcap _{i\in I}H_{i}$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
completely divisibility-closed normal subgroup	completely divisibility-closed and normal; equivalently, the quotient is torsion-free for any prime for which the whole group is divisible.			|FULL LIST, MORE INFO
kernel of a bihomomorphism		kernel of a bihomomorphism implies completely divisibility-closed
intersection of kernels of bihomomorphisms		intersection of kernels of bihomomorphisms implies completely divisibility-closed		|FULL LIST, MORE INFO
kernel of a multihomomorphism		kernel of a multihomomorphism implies completely divisibility-closed		|FULL LIST, MORE INFO
subgroup of finite group
subgroup of periodic group

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
divisibility-closed subgroup				|FULL LIST, MORE INFO
powering-invariant subgroup				|FULL LIST, MORE INFO