Local divisibility-closed subgroup

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Definition

Suppose $G$ is a group and $H$ is a subgroup of $G$. We say that $H$ is local divisibility-closed or local divisibility-invariant in $G$ if the following holds: for any $h\in H$ and any natural number $n$ such that the equation $x^{n}=h$ has a solution for $x$ in $G$, the equation $x^{n}=h$ has a solution for $x\in H$.

Relation with other properties

Conjunction with group properties for ambient group

• If the whole group is an abelian group, a more standard term for local divisibility-closed subgroup is pure subgroup.

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal Sylow subgroup
normal Hall subgroup
retract				|FULL LIST, MORE INFO
direct factor				|FULL LIST, MORE INFO
verbally closed subgroup				|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
local powering-invariant subgroup				|FULL LIST, MORE INFO
divisibility-closed subgroup			divisibility-closed not implies local divisibility-closed	|FULL LIST, MORE INFO
powering-invariant subgroup				|FULL LIST, MORE INFO

Incomparable properties

• Sylow subgroup: See Sylow not implies local divisibility-closed

Facts

• Derived subgroup not is local divisibility-closed in nilpotent group