Class-preserving automorphism - Revision history

Vipul: /* Stronger properties */

2023-09-01T14:42:33Z

Stronger properties

← Older revision		Revision as of 14:42, 1 September 2023
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	! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Weaker than::~~Inner~~ automorphism]] \|\| conjugation by an element \|\| [[inner implies class-preserving]] \|\| [[class-preserving not implies inner]] \|\| {{intermediate notions short\|class-preserving automorphism\|inner automorphism}}		\| [[Weaker than::inner automorphism]] \|\| conjugation by an element \|\| [[inner implies class-preserving]] \|\| [[class-preserving not implies inner]] \|\| {{intermediate notions short\|class-preserving automorphism\|inner automorphism}}
	\|-		\|-
	\| [[Weaker than::~~Locally~~ inner automorphism]] \|\| preserves conjugacy classes of finite tuples \|\| [[locally inner implies class-preserving]] \|\| [[class-preserving not implies locally inner]] \|\| {{intermediate notions short\|class-preserving automorphism\|locally inner automorphism}}		\| [[Weaker than::locally inner automorphism]] \|\| preserves conjugacy classes of finite tuples \|\| [[locally inner implies class-preserving]] \|\| [[class-preserving not implies locally inner]] \|\| {{intermediate notions short\|class-preserving automorphism\|locally inner automorphism}}
	\|}		\|}

Vipul: /* Weaker properties */

2015-05-19T22:27:06Z

Weaker properties

← Older revision		Revision as of 22:27, 19 May 2015
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Stronger than::automorphism that preserves conjugacy classes for a generating set]] \|\| there exists a [[generating set]] all of whose elements are sent to within their conjugacy class by the automorphism \|\| (obvious) \|\| [[preserves conjugacy classes for a generating set not implies class-preserving]] \|\| {{intermediate notions short\|automorphism that preserves conjugacy classes for a generating set\|class-preserving automorphism}}
	\|-		\|-
	\| [[Stronger than::IA-automorphism]] \|\| induces identity map on [[abelianization]] \|\| [[class-preserving implies IA]] \|\| [[IA not implies class-preserving]] \|\| {{intermediate notions short\|IA-automorphism\|class-preserving automorphism}}		\| [[Stronger than::IA-automorphism]] \|\| induces identity map on [[abelianization]] \|\| [[class-preserving implies IA]] \|\| [[IA not implies class-preserving]] \|\| {{intermediate notions short\|IA-automorphism\|class-preserving automorphism}}
	\|-		\|-
	\| [[Stronger than::~~Normal~~ automorphism]] \|\| sends each normal subgroup to itself \|\| [[class-preserving implies normal]] \|\| [[normal not implies class-preserving]] \|\| {{intermediate notions short\|normal automorphism\|class-preserving automorphism}}		\| [[Stronger than::normal automorphism]] \|\| sends each normal subgroup to itself \|\| [[class-preserving implies normal]] \|\| [[normal not implies class-preserving]] \|\| {{intermediate notions short\|normal automorphism\|class-preserving automorphism}}
	\|-		\|-
	\| [[Stronger than::~~Weakly~~ normal automorphism]] \|\| sends each normal subgroup to a subgroup of itself \|\| (via normal) \|\| (via normal) \|\| {{intermediate notions short\|weakly normal automorphism\|class-preserving automorphism}}		\| [[Stronger than::weakly normal automorphism]] \|\| sends each normal subgroup to a subgroup of itself \|\| (via normal) \|\| (via normal) \|\| {{intermediate notions short\|weakly normal automorphism\|class-preserving automorphism}}
	\|-		\|-
	\| [[Stronger than::~~Extended~~ class-preserving automorphism]] \|\| sends each element to conjugate or conjugate of inverse \|\| \|\| [[extended class-preserving not implies class-preserving]] \|\| {{intermediate notions short\|extended class-preserving automorphism\|class-preserving automorphism}}		\| [[Stronger than::extended class-preserving automorphism]] \|\| sends each element to conjugate or conjugate of inverse \|\| \|\| [[extended class-preserving not implies class-preserving]] \|\| {{intermediate notions short\|extended class-preserving automorphism\|class-preserving automorphism}}
	\|-		\|-
	\| [[Stronger than::~~Rational~~ class-preserving automorphism]] \|\| sends each element to conjugate of element generating same cyclic group \|\| \|\| \|\| {{intermediate notions short\|rational class-preserving automorphism\|class-preserving automorphism}}		\| [[Stronger than::rational class-preserving automorphism]] \|\| sends each element to conjugate of element generating same cyclic group \|\| \|\| \|\| {{intermediate notions short\|rational class-preserving automorphism\|class-preserving automorphism}}
	\|-		\|-
	\| [[Stronger than::~~Center~~-fixing automorphism]] \|\| fixes every element of [[center]] \|\| [[class-preserving implies center-fixing]] \|\|[[center-fixing not implies class-preserving]] \|\| {{intermediate notions short\|center-fixing automorphism\|class-preserving automorphism}}		\| [[Stronger than::center-fixing automorphism]] \|\| fixes every element of [[center]] \|\| [[class-preserving implies center-fixing]] \|\|[[center-fixing not implies class-preserving]] \|\| {{intermediate notions short\|center-fixing automorphism\|class-preserving automorphism}}
	\|}		\|}

Vipul: /* Weaker properties */

2015-05-19T05:43:18Z

Weaker properties

← Older revision		Revision as of 05:43, 19 May 2015
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	\| [[Stronger than::Normal automorphism]] \|\| sends each normal subgroup to itself \|\| [[class-preserving implies normal]] \|\| [[normal not implies class-preserving]] \|\| {{intermediate notions short\|normal automorphism\|class-preserving automorphism}}		\| [[Stronger than::Normal automorphism]] \|\| sends each normal subgroup to itself \|\| [[class-preserving implies normal]] \|\| [[normal not implies class-preserving]] \|\| {{intermediate notions short\|normal automorphism\|class-preserving automorphism}}
	\|-		\|-
	\| [[Stronger than::Weakly normal automorphism]] \|\| sends each normal subgroup to a subgroup to itself \|\| (via normal) \|\| (via normal) \|\| {{intermediate notions short\|weakly normal automorphism\|class-preserving automorphism}}		\| [[Stronger than::Weakly normal automorphism]] \|\| sends each normal subgroup to a subgroup of itself \|\| (via normal) \|\| (via normal) \|\| {{intermediate notions short\|weakly normal automorphism\|class-preserving automorphism}}
	\|-		\|-
	\| [[Stronger than::Extended class-preserving automorphism]] \|\| sends each element to conjugate or conjugate of inverse \|\| \|\| [[extended class-preserving not implies class-preserving]] \|\| {{intermediate notions short\|extended class-preserving automorphism\|class-preserving automorphism}}		\| [[Stronger than::Extended class-preserving automorphism]] \|\| sends each element to conjugate or conjugate of inverse \|\| \|\| [[extended class-preserving not implies class-preserving]] \|\| {{intermediate notions short\|extended class-preserving automorphism\|class-preserving automorphism}}
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	\| [[Stronger than::Center-fixing automorphism]] \|\| fixes every element of [[center]] \|\| [[class-preserving implies center-fixing]] \|\|[[center-fixing not implies class-preserving]] \|\| {{intermediate notions short\|center-fixing automorphism\|class-preserving automorphism}}		\| [[Stronger than::Center-fixing automorphism]] \|\| fixes every element of [[center]] \|\| [[class-preserving implies center-fixing]] \|\|[[center-fixing not implies class-preserving]] \|\| {{intermediate notions short\|center-fixing automorphism\|class-preserving automorphism}}
	\|}		\|}

	===Related properties===		===Related properties===

Vipul at 01:20, 11 January 2014

2014-01-11T01:20:18Z

← Older revision		Revision as of 01:20, 11 January 2014
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	{{automorphism property}}		{{automorphism property}}
	{{variationof\|inner automorphism}}		{{variationof\|inner automorphism}}

	~~==Origin==~~

	~~===Origin of the concept===~~

	The concept of class-preserving automorphism first took explicit shape when it was observed that there are automorphisms of groups that take each element to within its conjugacy class but are not [[inner automorphism\|inner]]. That is because there may not be a <em>single</em> element that serves uniformly as a ''conjugating candidate''.

	==Definition==		==Definition==

	~~===Symbol-free definition===~~		An [[defining ingredient::automorphism]] of a group is termed a '''class-preserving automorphism''' or '''class automorphism''' if it takes each element to within its [[defining ingredient::conjugacy class]]. In symbols, an automorphism <math>\sigma</math> of a group <math>G</math> is termed a '''class automorphism''' or '''class-preserving automorphism''' if for every <math>g</math> in <math>G</math>, there exists an element <math>h</math> such that <math>\sigma(g) = hgh^{-1}</math>. The choice of <math>h</math> may depend on <math>g</math>.

	An [[defining ingredient::automorphism]] of a group is termed a '''class-preserving automorphism''' or '''class automorphism''' if it takes each element to within its [[defining ingredient::conjugacy class]].

	~~===Definition with~~ symbols~~===~~

	An automorphism <math>\sigma</math> of a group <math>G</math> is termed a '''class automorphism''' or '''class-preserving automorphism''' if for every <math>g</math> in <math>G</math>, there exists an element <math>h</math> such that <math>\sigma(g) = hgh^{-1}</math>. The choice of <math>h</math> may depend on <math>g</math>.

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

Vipul: /* Relation with other properties */

2010-01-08T01:28:53Z

Relation with other properties

← Older revision		Revision as of 01:28, 8 January 2010
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	===Stronger properties===		===Stronger properties===

	* [[Weaker than::Inner automorphism]]: {{~~proofofstrictimplicationat~~\|[[~~Inner~~ implies class-preserving]]\|[[~~Class~~-preserving not implies inner]]}}		{\| class="sortable" border="1"
			! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Weaker than::Inner automorphism]] \|\| conjugation by an element \|\| [[inner implies class-preserving]] \|\| [[class-preserving not implies inner]] \|\| {{intermediate notions short\|class-preserving automorphism\|inner automorphism}}
			\|-
			\| [[Weaker than::Locally inner automorphism]] \|\| preserves conjugacy classes of finite tuples \|\| [[locally inner implies class-preserving]] \|\| [[class-preserving not implies locally inner]] \|\| {{intermediate notions short\|class-preserving automorphism\|locally inner automorphism}}
			\|}

	===Weaker properties===		===Weaker properties===

	* [[Stronger than::IA-automorphism]]~~: {{proofofstrictimplicationat~~\|[[~~Class~~-preserving implies IA]]\|[[IA not implies class-preserving]]}}		{\| class="sortable" border="1"
	* [[Stronger than::Normal automorphism]]~~: {{proofofstrictimplicationat~~\|[[~~Class~~-preserving implies normal]]\|[[~~Normal~~ not implies class-preserving]]}}		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* [[Stronger than::Weakly normal automorphism]]		\|-
	* [[Stronger than::Extended class-preserving automorphism]]		\| [[Stronger than::IA-automorphism]] \|\| induces identity map on [[abelianization]] \|\| [[class-preserving implies IA]] \|\| [[IA not implies class-preserving]] \|\| {{intermediate notions short\|IA-automorphism\|class-preserving automorphism}}
	* [[Stronger than::Rational class-preserving automorphism]]		\|-
	* [[Stronger than::Center-fixing automorphism]]~~: {{proofofstrictimplicationat~~\|[[~~Class~~-preserving implies center-fixing]]\|[[~~Center~~-fixing not implies class-preserving]]}}		\| [[Stronger than::Normal automorphism]] \|\| sends each normal subgroup to itself \|\| [[class-preserving implies normal]] \|\| [[normal not implies class-preserving]] \|\| {{intermediate notions short\|normal automorphism\|class-preserving automorphism}}
			\|-
			\| [[Stronger than::Weakly normal automorphism]] \|\| sends each normal subgroup to a subgroup to itself \|\| (via normal) \|\| (via normal) \|\| {{intermediate notions short\|weakly normal automorphism\|class-preserving automorphism}}
			\|-
			\| [[Stronger than::Extended class-preserving automorphism]] \|\| sends each element to conjugate or conjugate of inverse \|\| \|\| [[extended class-preserving not implies class-preserving]] \|\| {{intermediate notions short\|extended class-preserving automorphism\|class-preserving automorphism}}
			\|-
			\| [[Stronger than::Rational class-preserving automorphism]] \|\| sends each element to conjugate of element generating same cyclic group \|\| \|\| \|\| {{intermediate notions short\|rational class-preserving automorphism\|class-preserving automorphism}}
			\|-
			\| [[Stronger than::Center-fixing automorphism]] \|\| fixes every element of [[center]] \|\| [[class-preserving implies center-fixing]] \|\|[[center-fixing not implies class-preserving]] \|\| {{intermediate notions short\|center-fixing automorphism\|class-preserving automorphism}}
			\|}
	===Related properties===		===Related properties===

Vipul: /* Weaker properties */

2009-09-04T00:01:13Z

Weaker properties

← Older revision		Revision as of 00:01, 4 September 2009
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	* [[Stronger than::IA-automorphism]]: {{proofofstrictimplicationat\|[[Class-preserving implies IA]]\|[[IA not implies class-preserving]]}}		* [[Stronger than::IA-automorphism]]: {{proofofstrictimplicationat\|[[Class-preserving implies IA]]\|[[IA not implies class-preserving]]}}
	* [[Stronger than::Normal automorphism]]: {{proofofstrictimplicationat\|[[Class-preserving implies normal]]\|[[Normal not implies class-preserving]]}}		* [[Stronger than::Normal automorphism]]: {{proofofstrictimplicationat\|[[Class-preserving implies normal]]\|[[Normal not implies class-preserving]]}}
			* [[Stronger than::Weakly normal automorphism]]
	* [[Stronger than::Extended class-preserving automorphism]]		* [[Stronger than::Extended class-preserving automorphism]]
			* [[Stronger than::Rational class-preserving automorphism]]
	* [[Stronger than::Center-fixing automorphism]]: {{proofofstrictimplicationat\|[[Class-preserving implies center-fixing]]\|[[Center-fixing not implies class-preserving]]}}		* [[Stronger than::Center-fixing automorphism]]: {{proofofstrictimplicationat\|[[Class-preserving implies center-fixing]]\|[[Center-fixing not implies class-preserving]]}}

Vipul: /* Relation with other properties */

2009-09-01T23:50:09Z

Relation with other properties

← Older revision		Revision as of 23:50, 1 September 2009
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	* [[Stronger than::IA-automorphism]]: {{proofofstrictimplicationat\|[[Class-preserving implies IA]]\|[[IA not implies class-preserving]]}}		* [[Stronger than::IA-automorphism]]: {{proofofstrictimplicationat\|[[Class-preserving implies IA]]\|[[IA not implies class-preserving]]}}
	* [[Stronger than::Normal automorphism]]: {{proofofstrictimplicationat\|[[Class-preserving implies normal]]\|[[Normal not implies class-preserving]]}}		* [[Stronger than::Normal automorphism]]: {{proofofstrictimplicationat\|[[Class-preserving implies normal]]\|[[Normal not implies class-preserving]]}}
			* [[Stronger than::Extended class-preserving automorphism]]
	* [[Stronger than::Center-fixing automorphism]]: {{proofofstrictimplicationat\|[[Class-preserving implies center-fixing]]\|[[Center-fixing not implies class-preserving]]}}		* [[Stronger than::Center-fixing automorphism]]: {{proofofstrictimplicationat\|[[Class-preserving implies center-fixing]]\|[[Center-fixing not implies class-preserving]]}}

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	* [[Subgroup-conjugating automorphism]]: {{further\|[[Class-preserving not implies subgroup-conjugating]], [[Subgroup-conjugating not implies class-preserving]]}}		* [[Subgroup-conjugating automorphism]]: {{further\|[[Class-preserving not implies subgroup-conjugating]], [[Subgroup-conjugating not implies class-preserving]]}}
			* [[Class-inverting automorphism]]

	==Facts==		==Facts==

Vipul: /* Relation with other properties */

2009-09-01T23:45:41Z

Relation with other properties

← Older revision		Revision as of 23:45, 1 September 2009
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	* [[Weaker than::Inner automorphism]]: {{proofofstrictimplicationat\|[[Inner implies class-preserving]]\|[[Class-preserving not implies inner]]}}		* [[Weaker than::Inner automorphism]]: {{proofofstrictimplicationat\|[[Inner implies class-preserving]]\|[[Class-preserving not implies inner]]}}
	* [[Weaker than::Finite-extensible automorphism]]: {{proofofstrictimplicationat\|[[Finite-extensible implies class-preserving]]\|[[Cass-preserving not implies finite-extensible]]}}
	===Weaker properties===		===Weaker properties===

Vipul: /* Relation with other properties */

2009-02-03T03:16:04Z

Relation with other properties

← Older revision		Revision as of 03:16, 3 February 2009
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	* [[Weaker than::Inner automorphism]]: {{proofofstrictimplicationat\|[[Inner implies class-preserving]]\|[[Class-preserving not implies inner]]}}		* [[Weaker than::Inner automorphism]]: {{proofofstrictimplicationat\|[[Inner implies class-preserving]]\|[[Class-preserving not implies inner]]}}
			* [[Weaker than::Finite-extensible automorphism]]: {{proofofstrictimplicationat\|[[Finite-extensible implies class-preserving]]\|[[Cass-preserving not implies finite-extensible]]}}
	===Weaker properties===		===Weaker properties===

Vipul at 17:22, 26 September 2008

2008-09-26T17:22:55Z

← Older revision		Revision as of 17:22, 26 September 2008
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	* [[Subgroup-conjugating automorphism]]: {{further\|[[Class-preserving not implies subgroup-conjugating]], [[Subgroup-conjugating not implies class-preserving]]}}		* [[Subgroup-conjugating automorphism]]: {{further\|[[Class-preserving not implies subgroup-conjugating]], [[Subgroup-conjugating not implies class-preserving]]}}

			==Facts==

			* [[Class-preserving automorphism group of finite p-group is p-group]]

	==Metaproperties==		==Metaproperties==