Additive group of a field

Symbol-free definition
A nontrivial group is termed the additive group of a field if it satisfies the following equivalent conditions:


 * There exists a field whose additive group is isomorphic to the given group.
 * There exists a vector space over a field whose additive group is isomorphic to the given group.
 * The given group is an internal (restricted) direct product of copies of a cyclic group of prime order, or of the group of rational numbers.
 * It is abelian and characteristically simple.
 * It is abelian and it has no proper nontrivial fully invariant subgroup.
 * It is abelian, and its automorphism group is transitive on non-identity elements.

Stronger properties

 * Weaker than::Elementary abelian group (except the case of the trivial group, which is considered elementary abelian even though it is not the additive group of a field).

Weaker properties

 * Stronger than::Characteristically simple group
 * Stronger than::Abelian group