This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
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
- 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 .
|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|