Additive group of a field
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Definition
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.
Equivalence of definitions
Further information: Equivalence of definitions of additive group of a field
Relation with other properties
Stronger properties
- 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).