Additive group of a field: Difference between revisions
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{{basicdef in|field theory}} | |||
==Definition== | ==Definition== | ||
===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[group]] is termed the '''additive group of a field''' if it satisfies the following equivalent conditions: | 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 [[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. | * 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]]. | * 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 group| | * It is [[Abelian group|abelian]] and [[characteristically simple group|characteristically simple]]. | ||
* It is [[Abelian group| | * It is [[Abelian group|abelian]] and it has no proper nontrivial [[fully invariant subgroup]]. | ||
* It is | * It is abelian, and its [[group whose automorphism group is transitive on non-identity elements|automorphism group is transitive on non-identity elements]]. | ||
===Equivalence of definitions=== | ===Equivalence of definitions=== | ||
{{further|[[ | {{further|[[Equivalence of definitions of additive group of a field]]}} | ||
==Relation with other properties== | ==Relation with other properties== | ||
===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=== | ===Weaker properties=== | ||
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* [[Stronger than::Characteristically simple group]] | * [[Stronger than::Characteristically simple group]] | ||
* [[Stronger than::Abelian group]] | * [[Stronger than::Abelian group]] | ||
Latest revision as of 11:07, 9 December 2023
This article gives a basic definition in the following area: field theory
View other basic definitions in field theory |View terms related to field theory |View facts related to field theory
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).