Rationally powered group

Definition

A group $G$ is termed rationally powered or uniquely divisible if it satisfies the following equivalent conditions:

1. For every $u \in G$ and every natural number $n$, there is a unique $v \in G$ such that $v^n = u$.
2. $G$ is a powered group for all prime numbers.
3. For any integers $m,n$ with $n \ne 0$, and for any $g \in G$, there exists a unique $h \in G$ such that $g^m = h^n$.

More generally, we can talk of a powered group for a set of primes.

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Relation with other properties

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
torsion-free group no non-identity element of finite order obvious the group of integers is torsion-free but not rationally powered. Group in which every power map is injective|FULL LIST, MORE INFO
divisible group every element has at least one $n^{th}$ root for every $n$ obvious (rationally powered additionally guarantees uniqueness) the group of rational numbers modulo integers |FULL LIST, MORE INFO
powering-injective group no two different elements can have the same $n^{th}$ power. obvious the group of integers is powering-injective but not rationally powered. |FULL LIST, MORE INFO

Facts

• A nilpotent group is rationally powered iff it is divisible and torsion-free. This follows from the more general fact that in a nilpotent torsion-free group, if roots exist, they are unique.