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Contents 1 Definition 1.1 Symbol-free definition 1.2 Equivalence of definitions 2 Relation with other properties 2.1 Weaker properties

Definition

Symbol-free definition

A 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 FC-simple: it has no proper nontrivial fully characteristic subgroup.
• It is Abelian, and its automorphism group is transitive on non-identity elements.

Equivalence of definitions

Further information: Abelian and FC-simple implies additive group of a field

Relation with other properties

Weaker properties

• Characteristically simple group
• Abelian group
• Elementary Abelian group