# Divisibility-closed subgroup

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## 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 $H \le K \le G$ 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 Endomorphism image, Local divisibility-closed subgroup, Verbally closed subgroup|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) Completely divisibility-closed subgroup|FULL LIST, MORE INFO
kernel of a multihomomorphism (via completely divisibility-closed) Completely divisibility-closed subgroup|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 Divisibility-closed subgroup of abelian group|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