Divisible group

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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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A divisible group is a group G such that for any element g \in G, and any nonzero integer n, there exists h \in G such that h^n = g.

This notion is usually discussed for abelian groups divisible abelian group, where it coincides with the notion of an injective \mathbb{Z}-module. However, the notion is useful for more general kinds of groups, particularly for nilpotent groups.

Relation with other properties

Parametric versions

  • Divisible group for a set of primes is a group where it is possible to divide any group element by any prime in the specified set of primes. For a prime set \pi, a \pi-divisible group is a group G such that for any g \in G and p \in \pi, there exists x \in G such that x^p = g.

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
rationally powered group (also called uniquely divisible group) every element has a unique n^{th} root for every n. divisible not implies rationally powered |FULL LIST, MORE INFO
algebraically closed group every consistent system of equations and inequations has a solution in the group. Verbally complete group|FULL LIST, MORE INFO
verbally complete group every word map other than the identity word map is surjective. in a nontrivial divisible abelian group, the commutator word map is trivial and not surjective. |FULL LIST, MORE INFO

Conjunction with other properties

Conjunction Other component of conjunction Comments
divisible abelian group abelian group This is an injective object in the category of abelian groups (which is an abelian category).
divisible nilpotent group nilpotent group