# Additive group of a field

From Groupprops

## Contents

## 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).