Rationally powered group

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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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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

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.