Divisibility-closed subgroup

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Definition

A subgroup $H$ of a group $G$ is termed divisibility-closed or divisibility-invariant if it satisfies the following equivalent conditions:

1. For every prime $p$ such that $G$ is $p$-divisible (i.e., for every $g\in G$, there exists $x\in G$ such that $x^{p}=g$), $H$ is also $p$-divisible.
2. For every natural number $n$ such that $G$ is $n$-divisible (i.e., for every $g\in G$, there exists $x\in G$ such that $x^{n}=g$), $H$ is also $n$-divisible.

Note that we do not require that all the $n^{th}$ roots in $G$ of an element of $H$ must be in $H$.

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
transitive subgroup property	Yes	divisibility-closedness is transitive	If $H\leq K\leq G$ are groups such that $H$ is divisibility-closed in $K$ and $K$ is divisibility-closed in $G$, then $H$ is divisibility-closed in $G$.
trim subgroup property	Yes		In any group, the whole group and the trivial subgroup are divisibility-closed.
finite-intersection-closed subgroup property	No	divisibility-closedness is not finite-intersection-closed	It is possible to have a group $G$ and divisibility-closed subgroups $H,K$ of $G$ such that the intersection of subgroups $H\cap K$ is not divisibility-closed in $G$.
finite-join-closed subgroup property	No	divisibility-closedness is not finite-join-closed	It is possible to have a group $G$ and divisibility-closed subgroups $H,K$ of $G$ such that the join of subgroups $\langle H,K\rangle$ is not divisibility-closed in $G$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
direct factor	normal subgroup with normal complement	(via retract)
retract	has a normal complement			|FULL LIST, MORE INFO
subgroup of finite group				|FULL LIST, MORE INFO
local divisibility-closed subgroup	if a particular element of the subgroup has a $n^{th}$ root in the whole group, the element has a $n^{th}$ root in the subgroup.			|FULL LIST, MORE INFO
completely divisibility-closed subgroup	divisibility-closed, and all relevant roots lie inside the subgroup.			|FULL LIST, MORE INFO
kernel of a bihomomorphism	kernel of a bihomomorphism	(via completely divisibility-closed)		|FULL LIST, MORE INFO
kernel of a multihomomorphism		(via completely divisibility-closed)		|FULL LIST, MORE INFO
subgroup of finite index		(via completely divisibility-closed)		|FULL LIST, MORE INFO
pure subgroup	local divisibility-closed subgroup of abelian group	(via local divisibility-closed)	any example in a non-abelian group	|FULL LIST, MORE INFO
verbal subgroup of abelian group		verbal subgroup of abelian group implies divisibility-closed		|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
powering-invariant subgroup			powering-invariant not implies divisibility-closed

Satisfaction by subgroup-defining functions

Subgroup-defining function	Always divisibility-closed?	Proof	Stronger/weaker properties satisfied	If the group property is restricted
center	No	center not is divisibility-closed	weaker: powering-invariant subgroup (see center is powering-invariant and center is local powering-invariant)	in nilpotent group, center is a completely divisibility-closed subgroup (follows from upper central series members are completely divisibility-closed in nilpotent group)
members of the upper central series	No	based on failure for center	weaker: powering-invariant subgroup (see upper central series members are powering-invariant, upper central series members are quotient-powering-invariant)	in nilpotent group, they are divisibility-closed subgroups, and in fact are completely divisibility-closed subgroups (follows from upper central series members are completely divisibility-closed in nilpotent group)
derived subgroup	No	derived subgroup not is divisibility-closed		in nilpotent group, yes (see derived subgroup is divisibility-closed in nilpotent group, follows from equivalence of definitions of nilpotent group that is divisible for a set of primes)
members of the lower central series	No	based on failure for derived subgroup		in nilpotent group, they are divisibility-closed (see lower central series members are divisibility-closed in nilpotent group, follows from equivalence of definitions of nilpotent group that is divisible for a set of primes)
members of the derived series	No	based on failure for derived subgroup		in nilpotent group, they are divisibility-closed subgroups. This follows from the result for the derived subgroup. See derived series members are divisibility-closed in nilpotent group.
socle	No			Not true even in abelian groups, see monolith not is divisibility-closed in abelian group. However, the weaker property of being a powering-invariant subgroup is true in solvable groups: socle is powering-invariant in solvable group