Groupprops, The Group Properties Wiki (pre-alpha)

From Groupprops

Contents 1 Definition 1.1 Symbol-free definition 1.2 Equivalence of definitions 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties

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: Abelian and FC-simple implies 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).

Weaker properties

• Characteristically simple group
• Abelian group