Pronormal subgroup - Revision history

Vipul: /* Equivalent definitions in tabular format */

2014-11-10T04:06:33Z

Equivalent definitions in tabular format

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	\|1 \|\| Conjugates in whole group are conjugate in intermediate subgroups \|\| any [[defining ingredient::conjugate subgroups\|conjugate subgroup]] of the subgroup inside the whole group is also conjugate inside any intermediate subgroup. \|\| for any <math>g \in G</math> and any subgroup <math>K</math> of <math>G</math> containing both <math>H</math> and <math>H^g</math>, there exists <math>x \in K</math> such that <math>H^g = H^x</math>. \|\| for any <math>g \in G</math> and any subgroup <math>K</math> of <math>G</math> containing both <math>H</math> and <math>gHg^{-1}</math>, there exists <math>x \in K</math> such that <math>xHx^{-1} = gHg^{-1}</math>.		\|1 \|\| Conjugates in whole group are conjugate in intermediate subgroups \|\| any [[defining ingredient::conjugate subgroups\|conjugate subgroup]] of the subgroup inside the whole group is also conjugate inside any intermediate subgroup. \|\| for any <math>g \in G</math> and any subgroup <math>K</math> of <math>G</math> containing both <math>H</math> and <math>H^g</math>, there exists <math>x \in K</math> such that <math>H^g = H^x</math>. \|\| for any <math>g \in G</math> and any subgroup <math>K</math> of <math>G</math> containing both <math>H</math> and <math>gHg^{-1}</math>, there exists <math>x \in K</math> such that <math>xHx^{-1} = gHg^{-1}</math>.
	\|-		\|-
	\| 2 \|\| Conjugates in whole group are conjugate in join \|\| any [[defining ingredient::conjugate subgroups\|conjugate subgroup]] of the subgroup is conjugate to it inside the subgroup generated by the original subgroup and its conjugate. \|\| for any <math>g</math> in <math>G</math>, there exists <math>x \in \langle H,H^g \rangle</math> such that <math>H^g = H^x</math>. Here <math>H^g = g^{-1}Hg</math> and the angled braces are for the [[defining ingredient::join of subgroups]] (or subgroup generated). \|\| for any <math>g \in G</math>, there exists <math>x \in \langle H, gHg^{-1} \rangle</math> such that <math>gHg^{-1} = xHx^{-1}</math>.		\| 2 \|\| Conjugates in whole group are conjugate in join \|\| any [[defining ingredient::conjugate subgroups\|conjugate subgroup]] of the subgroup is conjugate to it inside the subgroup generated by the original subgroup and its conjugate. \|\| for any <math>g</math> in <math>G</math>, there exists <math>x \in \langle H,H^g \rangle</math> such that <math>H^g = H^x</math>. Here <math>H^g = g^{-1}Hg</math> and the angled braces are for the [[defining ingredient::join of subgroups]] (i.e., the subgroup generated). \|\| for any <math>g \in G</math>, there exists <math>x \in \langle H, gHg^{-1} \rangle</math> such that <math>gHg^{-1} = xHx^{-1}</math>.
	\|}		\|}

Vipul: /* Definition */

2014-11-10T04:06:07Z

Definition

← Older revision		Revision as of 04:06, 10 November 2014
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	\|1 \|\| Conjugates in whole group are conjugate in intermediate subgroups \|\| any [[defining ingredient::conjugate subgroups\|conjugate subgroup]] of the subgroup inside the whole group is also conjugate inside any intermediate subgroup. \|\| for any <math>g \in G</math> and any subgroup <math>K</math> of <math>G</math> containing both <math>H</math> and <math>H^g</math>, there exists <math>x \in K</math> such that <math>H^g = H^x</math>. \|\| for any <math>g \in G</math> and any subgroup <math>K</math> of <math>G</math> containing both <math>H</math> and <math>gHg^{-1}</math>, there exists <math>x \in K</math> such that <math>xHx^{-1} = gHg^{-1}</math>.		\|1 \|\| Conjugates in whole group are conjugate in intermediate subgroups \|\| any [[defining ingredient::conjugate subgroups\|conjugate subgroup]] of the subgroup inside the whole group is also conjugate inside any intermediate subgroup. \|\| for any <math>g \in G</math> and any subgroup <math>K</math> of <math>G</math> containing both <math>H</math> and <math>H^g</math>, there exists <math>x \in K</math> such that <math>H^g = H^x</math>. \|\| for any <math>g \in G</math> and any subgroup <math>K</math> of <math>G</math> containing both <math>H</math> and <math>gHg^{-1}</math>, there exists <math>x \in K</math> such that <math>xHx^{-1} = gHg^{-1}</math>.
	\|-		\|-
	\| 2 \|\| Conjugates in whole group are conjugate in join \|\| any [[defining ingredient::conjugate subgroups\|conjugate subgroup]] of the subgroup is conjugate to it inside the subgroup generated by the original subgroup and its conjugate. \|\| for any <math>g</math> in <math>G</math>, there exists <math>x \in \langle H,H^g \rangle</math> such that <math>H^g = H^x</math>. Here <math>{}^gH = g^{-1}Hg</math> and the angled braces are for the [[defining ingredient::join of subgroups]] (or subgroup generated). \|\| for any <math>g \in G</math>, there exists <math>x \in \langle H, gHg^{-1} \rangle</math> such that <math>gHg^{-1} = xHx^{-1}</math>.		\| 2 \|\| Conjugates in whole group are conjugate in join \|\| any [[defining ingredient::conjugate subgroups\|conjugate subgroup]] of the subgroup is conjugate to it inside the subgroup generated by the original subgroup and its conjugate. \|\| for any <math>g</math> in <math>G</math>, there exists <math>x \in \langle H,H^g \rangle</math> such that <math>H^g = H^x</math>. Here <math>H^g = g^{-1}Hg</math> and the angled braces are for the [[defining ingredient::join of subgroups]] (or subgroup generated). \|\| for any <math>g \in G</math>, there exists <math>x \in \langle H, gHg^{-1} \rangle</math> such that <math>gHg^{-1} = xHx^{-1}</math>.
	\|}		\|}

Vipul at 22:45, 11 March 2013

2013-03-11T22:45:11Z

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

	The two definitions are equivalent because being conjugate inside the ''smallest'' possible intermediate subgroup, ~~viz~~ <math>\langle H,H^g \rangle</math>, implies being conjugate in any intermediate subgroup.		The two definitions are equivalent because being conjugate inside the ''smallest'' possible intermediate subgroup, i.e., <math>\langle H,H^g \rangle</math>, implies being conjugate in any intermediate subgroup.

	{{stdnonbasicdef}}		{{stdnonbasicdef}}

Vipul at 03:00, 17 December 2011

2011-12-17T03:00:15Z

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			[[importance rank::3\| ]]
	==Definition==		==Definition==

Vipul at 23:17, 16 September 2011

2011-09-16T23:17:16Z

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	Under the right action convention, the right action of <math>g</math> on <math>H</math> by conjugation gives the subgroup <math>H^g = g^{-1}Hg</math>. Under the left action convention, the left action of <math>g</math> on <math>H</math> by conjugation gives the subgroup <math>{}^gH = gHg^{-1}</math>. Note that although the actions differ, the notion of being [[conjugate subgroups]] inside an intermediate subgroup remains unchanged. This can be explained by the fact that [[every group is isomorphic to its opposite group\|the inverse map reverses the roles of left and right]] while preserving subgroups.		Under the right action convention, the right action of <math>g</math> on <math>H</math> by conjugation gives the subgroup <math>H^g = g^{-1}Hg</math>. Under the left action convention, the left action of <math>g</math> on <math>H</math> by conjugation gives the subgroup <math>{}^gH = gHg^{-1}</math>. Note that although the actions differ, the notion of being [[conjugate subgroups]] inside an intermediate subgroup remains unchanged. This can be explained by the fact that [[every group is isomorphic to its opposite group\|the inverse map reverses the roles of left and right]] while preserving subgroups.

			{{tabular definition format}}

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

Vipul at 01:14, 22 June 2011

2011-06-22T01:14:51Z

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	~~{{DISPLAYTITLE:<span style="display">{{FULLPAGENAME}}{{#widget:Facebook Like Button\|name = {{FULLPAGENAME}}\|ref = top}}</span>}}~~
	==Definition==		==Definition==

Vipul at 01:13, 22 June 2011

2011-06-22T01:13:52Z

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	{{DISPLAYTITLE:{{FULLPAGENAME}}{{#widget:Facebook Like Button\|name = {{FULLPAGENAME}}\|ref = top}}}}		{{DISPLAYTITLE:<span style="display">{{FULLPAGENAME}}{{#widget:Facebook Like Button\|name = {{FULLPAGENAME}}\|ref = top}}</span>}}
	==Definition==		==Definition==

Vipul at 01:13, 22 June 2011

2011-06-22T01:13:14Z

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	{{DISPLAYTITLE:~~<span style="display:none">~~{{FULLPAGENAME}}{{#widget:Facebook Like Button\|name = {{FULLPAGENAME}}\|ref = top}}~~</span>~~}}		{{DISPLAYTITLE:{{FULLPAGENAME}}{{#widget:Facebook Like Button\|name = {{FULLPAGENAME}}\|ref = top}}}}
	==Definition==		==Definition==

Vipul at 01:12, 22 June 2011

2011-06-22T01:12:27Z

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			{{DISPLAYTITLE:<span style="display:none">{{FULLPAGENAME}}{{#widget:Facebook Like Button\|name = {{FULLPAGENAME}}\|ref = top}}</span>}}
	==Definition==		==Definition==

	{{quick phrase\|conjugates in whole group are conjugate in intermediate subgroups, conjugates in whole group are conjugate in join~~{{#widget:Facebook Like Button\|name = {{FULLPAGENAME}}\|ref = top}}~~}}		{{quick phrase\|conjugates in whole group are conjugate in intermediate subgroups, conjugates in whole group are conjugate in join}}
	===Equivalent definitions in tabular format===		===Equivalent definitions in tabular format===

Vipul at 01:01, 22 June 2011

2011-06-22T01:01:17Z

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	* {{paperlink\|Hallonsystemnormalizers}}		* {{paperlink\|Hallonsystemnormalizers}}
	* {{paperlink\|Peng-pronormal}}		* {{paperlink\|Peng-pronormal}}

	~~==External links==~~

	~~{{searchbox\|pronormal}}~~
	~~===Definition links===~~

	* {{sor\|p/p075410}}
	* {{planetmath\|Pronormality}}
	* {{wp\|Pronormal subgroup}}

← Older revision		Revision as of 01:01, 22 June 2011
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	* {{paperlink\|Hallonsystemnormalizers}}		* {{paperlink\|Hallonsystemnormalizers}}
	* {{paperlink\|Peng-pronormal}}		* {{paperlink\|Peng-pronormal}}

	~~==External links==~~

	~~{{searchbox\|pronormal}}~~
	~~===Definition links===~~

	* {{sor\|p/p075410}}
	* {{planetmath\|Pronormality}}
	* {{wp\|Pronormal subgroup}}