Additive group of a field - Revision history

R-a-jones at 11:07, 9 December 2023

2023-12-09T11:07:06Z

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			{{basicdef in\|field theory}}

	==Definition==		==Definition==

Vipul: /* Equivalence of definitions */

2011-09-12T16:57:08Z

Equivalence of definitions

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	===Equivalence of definitions===		===Equivalence of definitions===

	{{further\|[[~~Abelian and FC-simple implies~~ additive group of a field]]}}		{{further\|[[Equivalence of definitions of additive group of a field]]}}

	==Relation with other properties==		==Relation with other properties==

Vipul at 23:53, 3 September 2009

2009-09-03T23:53:37Z

← Older revision		Revision as of 23:53, 3 September 2009
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	===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.
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	==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]]
	* [[Stronger than::Elementary Abelian group]]

Vipul: /* Definition */

2009-09-02T21:11:45Z

Definition

← Older revision		Revision as of 21:11, 2 September 2009
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	* 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\|~~Abelian~~]] and [[characteristically simple group\|characteristically simple]].		* It is [[Abelian group\|abelian]] and [[characteristically simple group\|characteristically simple]].
	* It is [[Abelian group\|~~Abelian~~]] and ~~[[FC-simple group\|FC-simple]]:~~ it has no proper nontrivial [[fully ~~characteristic~~ subgroup]].		* It is [[Abelian group\|abelian]] and it has no proper nontrivial [[fully invariant subgroup]].
	* It is ~~Abelian~~, and its [[group whose automorphism group is transitive on non-identity elements\|automorphism group is transitive on non-identity elements]].		* 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\|[[Abelian and FC-simple implies additive group of a field]]}}		{{further\|[[Abelian and FC-simple implies additive group of a field]]}}

	==Relation with other properties==		==Relation with other properties==

Vipul at 20:47, 12 August 2008

2008-08-12T20:47:25Z

← Older revision		Revision as of 20:47, 12 August 2008
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	* 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\|Abelian]] and [[characteristically simple group\|characteristically simple]].		* It is [[Abelian group\|Abelian]] and [[characteristically simple group\|characteristically simple]].
	* It is [[Abelian group\|Abelian]] and FC-simple group\|FC-simple]]: it has no proper nontrivial [[fully characteristic subgroup]].		* It is [[Abelian group\|Abelian]] and [[FC-simple group\|FC-simple]]: it has no proper nontrivial [[fully characteristic subgroup]].
	* It is Abelian, and its [[group whose automorphism group is transitive on non-identity elements\|automorphism group is transitive on non-identity elements]].		* It is Abelian, and its [[group whose automorphism group is transitive on non-identity elements\|automorphism group is transitive on non-identity elements]].

Vipul at 20:58, 9 August 2008

2008-08-09T20:58:03Z

← Older revision		Revision as of 20:58, 9 August 2008
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	* 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\|Abelian]] and [[characteristically simple group\|characteristically simple]].		* It is [[Abelian group\|Abelian]] and [[characteristically simple group\|characteristically simple]].
			* It is [[Abelian group\|Abelian]] and FC-simple group\|FC-simple]]: it has no proper nontrivial [[fully characteristic subgroup]].
	* It is Abelian, and its [[group whose automorphism group is transitive on non-identity elements\|automorphism group is transitive on non-identity elements]].		* 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\|[[Abelian and ~~characteristically~~ simple implies additive group of a field]]}}		{{further\|[[Abelian and FC-simple implies additive group of a field]]}}
	==Relation with other properties==		==Relation with other properties==

Vipul at 20:51, 9 August 2008

2008-08-09T20:51:50Z

← Older revision		Revision as of 20:51, 9 August 2008
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	==Definition==		==Definition==

			===Symbol-free definition===

	A [[group]] is termed the '''additive group of a field''' if it satisfies the following equivalent conditions:		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 [[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\|Abelian]] and [[characteristically simple group\|characteristically simple]].
			* 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===

			{{further\|[[Abelian and characteristically simple implies additive group of a field]]}}
	==Relation with other properties==		==Relation with other properties==

Vipul: New page: ==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 isomorphi...

2008-05-19T15:41:08Z

New page: ==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 isomorphi...

New page

==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]]

==Relation with other properties==

===Weaker properties===

* [[Stronger than::Characteristically simple group]]
* [[Stronger than::Abelian group]]
* [[Stronger than::Elementary Abelian group]]