Polynormal subgroup - Revision history

Vipul: /* Testing */

2008-11-22T22:52:13Z

Testing

← Older revision		Revision as of 22:52, 22 November 2008
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	==Testing==		==Testing==

	{{GAP code for ~~sgp~~\|test = IsPolynormal}}		{{GAP code for subgroup property\|test = IsPolynormal}}

	==References==		==References==

Vipul: /* References */

2008-10-30T22:01:58Z

References

← Older revision		Revision as of 22:01, 30 October 2008
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	In fact, an arbitrary, possibly empty, join of polynormal subgroups is polynormal. {{proofat\|[[Polynormality is strongly join-closed]]}}		In fact, an arbitrary, possibly empty, join of polynormal subgroups is polynormal. {{proofat\|[[Polynormality is strongly join-closed]]}}
			==Testing==

			{{GAP code for sgp\|test = IsPolynormal}}
	==References==		==References==

Vipul: /* Relation with other properties */

2008-10-07T18:48:38Z

Relation with other properties

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	* [[Weaker than::Strongly polynormal subgroup]]		* [[Weaker than::Strongly polynormal subgroup]]
	* [[Weaker than::Sylow subgroup]] in a [[finite group]]		* [[Weaker than::Sylow subgroup]] in a [[finite group]]
	* [[Weaker than::Intermediately contranormal subgroup]]

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

Vipul at 13:04, 25 September 2008

2008-09-25T13:04:34Z

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	{{intsubcondn}}		{{intsubcondn}}

	If <math>H</math> is polynormal in <math>G</math>, <math>H</math> is also polynormal in any intermediate subgroup <math>K</math>. {{~~further~~\|[[Polynormality satisfies intermediate subgroup condition]]}}		If <math>H</math> is polynormal in <math>G</math>, <math>H</math> is also polynormal in any intermediate subgroup <math>K</math>. {{proofat\|[[Polynormality satisfies intermediate subgroup condition]]}}

	{{trim}}		{{trim}}
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	{{join-closed}}		{{join-closed}}

	In fact, an arbitrary, possibly empty, join of polynormal subgroups is polynormal. {{~~further~~\|[[Polynormality is strongly join-closed]]}}		In fact, an arbitrary, possibly empty, join of polynormal subgroups is polynormal. {{proofat\|[[Polynormality is strongly join-closed]]}}
	==References==		==References==

Vipul at 13:04, 25 September 2008

2008-09-25T13:04:11Z

← Older revision		Revision as of 13:04, 25 September 2008
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	{{intsubcondn}}		{{intsubcondn}}

	If <math>H</math> is polynormal in <math>G</math>, <math>H</math> is also polynormal in any intermediate subgroup <math>K</math>.		If <math>H</math> is polynormal in <math>G</math>, <math>H</math> is also polynormal in any intermediate subgroup <math>K</math>. {{further\|[[Polynormality satisfies intermediate subgroup condition]]}}

	{{trim}}		{{trim}}
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	The whole group and the trivial subgroup are polynormal; in fact they are [[normal subgroup\|normal]].		The whole group and the trivial subgroup are polynormal; in fact they are [[normal subgroup\|normal]].

			{{join-closed}}

			In fact, an arbitrary, possibly empty, join of polynormal subgroups is polynormal. {{further\|[[Polynormality is strongly join-closed]]}}
	==References==		==References==

Vipul at 20:34, 24 September 2008

2008-09-24T20:34:13Z

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

	A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''polynormal''' if given any <math>g \in G</math>, ~~there exists a~~ <math>~~x \in~~ H~~^{\langle g \rangle}~~</math> ~~such that~~ <math>~~H^{\langle x\rangle} =~~ H^{\langle g \rangle}</math>. ~~Here <math>H^{\langle g \rangle}</math> denotes~~ the ~~smallest subgroup~~ of ~~<math>G</math> containing~~ <math>H</math>~~, which is closed~~ under conjugation by <math>g~~</math> and <math>g^{-1}~~</math>.		A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''polynormal''' if given any <math>g \in G</math>, <math>H</math> is a [[contranormal subgroup]] in the subgroup <math>H^{\langle g \rangle}</math>, i.e., the closure of <math>H</math> under the action by conjugation of the cyclic subgroup generated by <math>g</math>.

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

	* [[Normal subgroup]]		* [[Weaker than::Normal subgroup]]
	* [[Maximal subgroup]]		* [[Weaker than::Maximal subgroup]]
	* [[Abnormal subgroup]]		* [[Weaker than::Abnormal subgroup]]
	* [[Pronormal subgroup]]		* [[Weaker than::Pronormal subgroup]]
	* [[Weakly abnormal subgroup]]		* [[Weaker than::Weakly abnormal subgroup]]
	* [[Weakly pronormal subgroup]]		* [[Weaker than::Weakly pronormal subgroup]]
	* [[Paranormal subgroup]]		* [[Weaker than::Strongly paranormal subgroup]]
	* [[Sylow subgroup]] in a [[finite group]]		* [[Weaker than::Paranormal subgroup]]
			* [[Weaker than::Strongly polynormal subgroup]]
			* [[Weaker than::Sylow subgroup]] in a [[finite group]]
			* [[Weaker than::Intermediately contranormal subgroup]]

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

	* [[Fan subgroup]]		* [[Stronger than::Fan subgroup]]
	* [[Intermediately subnormal-to-normal subgroup]]: {{proofat\|[[Polynormal implies intermediately subnormal-to-normal]]}}		* [[Stronger than::Intermediately subnormal-to-normal subgroup]]: {{proofat\|[[Polynormal implies intermediately subnormal-to-normal]]}}

	==Metaproperties==		==Metaproperties==

Vipul: 11 revisions

2008-05-08T00:01:21Z

11 revisions

← Older revision	Revision as of 00:01, 8 May 2008
(No difference)

Vipul: /* Definition with symbols */

2008-03-27T16:15:20Z

Definition with symbols

← Older revision		Revision as of 16:15, 27 March 2008
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	===Definition with symbols===		===Definition with symbols===

	A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''polynormal''' if given any <math>g \in G</math>, there exists a <math>x \in H^{\langle g \rangle}</math> such that <math>H^{\langle x\rangle} = H^{\langle g \rangle}</math>. Here <math>H^{\langle g \rangle}</math> denotes the smallest subgroup of <math>G</math> containing <math>H</math>, which is closed under conjugation by <math>g</math>.		A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''polynormal''' if given any <math>g \in G</math>, there exists a <math>x \in H^{\langle g \rangle}</math> such that <math>H^{\langle x\rangle} = H^{\langle g \rangle}</math>. Here <math>H^{\langle g \rangle}</math> denotes the smallest subgroup of <math>G</math> containing <math>H</math>, which is closed under conjugation by <math>g</math> and <math>g^{-1}</math>.

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

Vipul: /* Definition */

2008-02-15T19:17:53Z

Definition

← Older revision		Revision as of 19:17, 15 February 2008
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	A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''polynormal''' if given any <math>g \in G</math>, there exists a <math>x \in H^{\langle g \rangle}</math> such that <math>H^{\langle x\rangle} = H^{\langle g \rangle}</math>. Here <math>H^{\langle g \rangle}</math> denotes the smallest subgroup of <math>G</math> containing <math>H</math>, which is closed under conjugation by <math>g</math>.		A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''polynormal''' if given any <math>g \in G</math>, there exists a <math>x \in H^{\langle g \rangle}</math> such that <math>H^{\langle x\rangle} = H^{\langle g \rangle}</math>. Here <math>H^{\langle g \rangle}</math> denotes the smallest subgroup of <math>G</math> containing <math>H</math>, which is closed under conjugation by <math>g</math>.

	~~Here <math>H^g = gHg^{-1}</math> denotes the conjugate of <math>H</math> by <math>g</math> and the angled braces denote the subgroup generated (i.e. [[join of subgroups]]).~~

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

Vipul: /* Definition */

2008-02-15T19:17:35Z

Definition

← Older revision		Revision as of 19:17, 15 February 2008
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	===Definition with symbols===		===Definition with symbols===

	A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''polynormal''' if given any <math>g \in G</math>, there exists a <math>x \in H^{\langle g \rangle}</math> such that <math>H^{<x>} = H^{<g>}</math>. Here <math>H^{\langle g \rangle}</math> denotes the smallest subgroup of <math>G</math> containing <math>H</math>, which is closed under conjugation by <math>g</math>.		A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''polynormal''' if given any <math>g \in G</math>, there exists a <math>x \in H^{\langle g \rangle}</math> such that <math>H^{\langle x\rangle} = H^{\langle g \rangle}</math>. Here <math>H^{\langle g \rangle}</math> denotes the smallest subgroup of <math>G</math> containing <math>H</math>, which is closed under conjugation by <math>g</math>.

			Here <math>H^g = gHg^{-1}</math> denotes the conjugate of <math>H</math> by <math>g</math> and the angled braces denote the subgroup generated (i.e. [[join of subgroups]]).

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