Divisible group
From Groupprops
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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Contents
Definition
A divisible group is a group such that for any element , and any nonzero integer , there exists such that .
This notion is usually discussed for abelian groups divisible abelian group, where it coincides with the notion of an injective -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 , a -divisible group is a group such that for any and , there exists such that .
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 root for every . | 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 |