Normal not implies characteristic - Revision history

Vipul: /* Implementation of the generic example */

2021-02-16T17:08:21Z

Implementation of the generic example

← Older revision		Revision as of 17:08, 16 February 2021
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	===Implementation of the generic example===		===Implementation of the generic example===

	Before using this generic example, you need to define <math>G</math> for GAP, choosing any nontrivial group (double semicolons have been used here to suppress GAP's output for the first three commands, which depends on the specific choice of <math>G</math> -- you can use single semicolons instead).		Before using this generic example, you need to define <math>H</math> for GAP, choosing any nontrivial group (double semicolons have been used here to suppress GAP's output for the first three commands, which depends on the specific choice of <math>H</math> -- you can use single semicolons instead).

	<pre>gap> G := DirectProduct(H,H);;		<pre>gap> G := DirectProduct(H,H);;

2013-07-26T16:52:03Z

Stronger formulations using the same examples

← Older revision		Revision as of 16:52, 26 July 2013
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	===Stronger formulations using the same examples===		===Stronger formulations using the same examples===

	* [[Normal not implies characteristic in the collection of groups satisfying a nontrivial finite direct product-closed group property]]		* [[Normal not implies characteristic in the collection of all groups satisfying a nontrivial finite direct product-closed group property]]
	* [[Every nontrivial normal subgroup is potentially normal-and-not-characteristic]]		* [[Every nontrivial normal subgroup is potentially normal-and-not-characteristic]]
	* [[Conjunction of normality with a nontrivially satisfied group property not implies characteristic]]		* [[Conjunction of normality with a nontrivially satisfied group property not implies characteristic]]

2013-07-26T16:51:32Z

Stronger formulations using the same examples

← Older revision		Revision as of 16:51, 26 July 2013
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	===Stronger formulations using the same examples===		===Stronger formulations using the same examples===

	* [[Normal not implies characteristic in ~~any~~ collection of groups satisfying a nontrivial finite direct product-closed group property]]		* [[Normal not implies characteristic in the collection of groups satisfying a nontrivial finite direct product-closed group property]]
	* [[Every nontrivial normal subgroup is potentially normal-and-not-characteristic]]		* [[Every nontrivial normal subgroup is potentially normal-and-not-characteristic]]
	* [[Conjunction of normality with a nontrivially satisfied group property not implies characteristic]]		* [[Conjunction of normality with a nontrivially satisfied group property not implies characteristic]]

2013-07-26T16:51:13Z

Stronger formulations using the same examples

← Older revision		Revision as of 16:51, 26 July 2013
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	===Stronger formulations using the same examples===		===Stronger formulations using the same examples===

	* [[Normal not implies characteristic in any ~~nontrivial subquasivariety of the quasivariety~~ of groups]]		* [[Normal not implies characteristic in any collection of groups satisfying a nontrivial finite direct product-closed group property]]
	* [[Every nontrivial normal subgroup is potentially normal-and-not-characteristic]]		* [[Every nontrivial normal subgroup is potentially normal-and-not-characteristic]]
	* [[Conjunction of normality with a nontrivially satisfied group property not implies characteristic]]		* [[Conjunction of normality with a nontrivially satisfied group property not implies characteristic]]

2013-01-20T02:31:58Z

Initial examples

← Older revision		Revision as of 02:31, 20 January 2013
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	===Initial examples===		===Initial examples===

	{{strictness examples for ~~group~~ property sorted by importance rank\|normal subgroup\|characteristic subgroup}}		{{strictness examples for subgroup property sorted by importance rank\|normal subgroup\|characteristic subgroup}}

	==GAP implementation==		==GAP implementation==

2013-01-20T02:31:04Z

Initial examples

← Older revision		Revision as of 02:31, 20 January 2013
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	===Initial examples===		===Initial examples===

	{{strictness examples sorted by importance rank\|normal subgroup\|characteristic subgroup}}		{{strictness examples for group property sorted by importance rank\|normal subgroup\|characteristic subgroup}}

	==GAP implementation==		==GAP implementation==

2013-01-20T02:30:37Z

← Older revision		Revision as of 02:30, 20 January 2013
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	==Statement==		==Statement==

	It is possible to have a [[group]] <math>G</math> and a [[~~fact about::~~normal subgroup]] <math>H</math> of <math>G</math> that is not a [[characteristic subgroup]] of <math>G</math>.		It is possible to have a [[group]] <math>G</math> and a [[normal subgroup]] <math>H</math> of <math>G</math> that is not a [[characteristic subgroup]] of <math>G</math>.

	==Related facts==		==Related facts==
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	* When <math>H</math> is [[cyclic group:Z2\|the cyclic group of order two]], <math>G</math> is the [[particular example::Klein four-group]]. In particular, this gives a counterexample where the ambient group is an [[abelian group]]. More generally, we can start with any nontrivial abelian group <math>H</math>.		* When <math>H</math> is [[cyclic group:Z2\|the cyclic group of order two]], <math>G</math> is the [[particular example::Klein four-group]]. In particular, this gives a counterexample where the ambient group is an [[abelian group]]. More generally, we can start with any nontrivial abelian group <math>H</math>.

			===Initial examples===

			{{strictness examples sorted by importance rank\|normal subgroup\|characteristic subgroup}}

	==GAP implementation==		==GAP implementation==

2011-09-16T22:41:38Z

← Older revision		Revision as of 22:41, 16 September 2011
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	{{subgroup property non-implication\|		{{subgroup property non-implication\|
	stronger = normal subgroup\|		stronger = normal subgroup\|
	weaker = characteristic subgroup}}		weaker = characteristic subgroup\|
			stronger relevance = 1\|
			weaker relevance = 1}}
	[[difficulty level::2\| ]]		[[difficulty level::2\| ]]
	==Statement==		==Statement==

2011-06-18T21:54:03Z

Statement

← Older revision		Revision as of 21:54, 18 June 2011
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	==Statement==		==Statement==

	It is possible to have a [[group]] <math>G</math> and a [[fact about::normal subgroup]] <math>H</math> of <math>G</math> that is ~~a normal subgroup of <math>G</math> but~~ not a [[characteristic subgroup]] of <math>G</math>.		It is possible to have a [[group]] <math>G</math> and a [[fact about::normal subgroup]] <math>H</math> of <math>G</math> that is not a [[characteristic subgroup]] of <math>G</math>.

	==Related facts==		==Related facts==

2011-06-18T21:53:48Z

Statement

← Older revision		Revision as of 21:53, 18 June 2011
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	==Statement==		==Statement==

	~~===Verbal statement===~~		It is possible to have a [[group]] <math>G</math> and a [[fact about::normal subgroup]] <math>H</math> of <math>G</math> that is a normal subgroup of <math>G</math> but not a [[characteristic subgroup]] of <math>G</math>.
	~~A [[fact about::normal subgroup]] of a group need not be a [[fact about::characteristic subgroup]].~~

	~~===Statement with symbols===~~

	It is possible to have a [[group]] <math>G</math> and a [[normal subgroup]] <math>H</math> of <math>G</math> that is a normal subgroup of <math>G</math> but not a [[characteristic subgroup]] of <math>G</math>.

	==Related facts==		==Related facts==