Permutable subgroup - Revision history

Vipul at 03:01, 17 December 2011

2011-12-17T03:01:01Z

← Older revision		Revision as of 03:01, 17 December 2011
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			[[importance rank::3\| ]]
	{{semibasicdef}}		{{semibasicdef}}
	{{subgroup property}}		{{subgroup property}}

Vipul at 17:14, 28 June 2009

2009-06-28T17:14:23Z

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	{{subgroup property}}		{{subgroup property}}

	{{~~variationof~~\|normality}}		{{variation of\|normality}}

	==History==		==History==

Vipul at 18:40, 2 April 2009

2009-04-02T18:40:12Z

← Older revision		Revision as of 18:40, 2 April 2009
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	It is possible to have a group <math>G</math>, a subgroup <math>H</math> of <math>G</math>, and intermediate subgroups <math>K_1</math> and <math>K_2</math> such that <math>H</math> is permutable in both <math>K_1</math> and <math>K_2</math> but <math>H</math> is ''not'' permutable in <math>\langle K_1, K_2 \rangle</math>. {{proofat\|[[Permutability is not upper join-closed]]}}		It is possible to have a group <math>G</math>, a subgroup <math>H</math> of <math>G</math>, and intermediate subgroups <math>K_1</math> and <math>K_2</math> such that <math>H</math> is permutable in both <math>K_1</math> and <math>K_2</math> but <math>H</math> is ''not'' permutable in <math>\langle K_1, K_2 \rangle</math>. {{proofat\|[[Permutability is not upper join-closed]]}}

			==Effect of property operators==

			{{applyingoperatorgives\|simple group operator\|simple group}}

			A group is a [[simple group]] if and only if it has no proper nontrivial permutable subgroup. {{proofat\|[[Simple iff no proper nontrivial permutable subgroup]]}}

	==Testing==		==Testing==

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	* {{booklink-defined\|Schmidt}}, Page 43, ''Modular and permutable subgroups''		* {{booklink-defined\|Schmidt}}, Page 43, ''Modular and permutable subgroups''

	~~==External links==~~
	~~{{searchbox\|"permutable+subgroup"OR"quasinormal+subgroup"}}~~
	~~===Definition links===~~
	* {{wp\|Permutable_subgroup}}
	* {{planetmath\|Quasinormal}}

	===Articles===		===Articles===

	* [http://www.maths.tcd.ie/pub/ims/bull56/GiG5612.pdf Old, recent and new results on quasinormal subgroups]		* [http://www.maths.tcd.ie/pub/ims/bull56/GiG5612.pdf Old, recent and new results on quasinormal subgroups]

Vipul: /* Relation with other properties */

2009-03-30T16:46:42Z

Relation with other properties

← Older revision		Revision as of 16:46, 30 March 2009
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	===Weaker properties===		===Weaker properties===

	* [[Stronger than in finite::Nilpotent quotient-by-core subgroup]] (when we are working with a [[finite group]])
	* [[Stronger than::Ascendant subgroup]]		* [[Stronger than::Ascendant subgroup]]
	* [[Stronger than in finite::Subnormal subgroup]] (when we are working with a [[finite group]])
	* [[Stronger than::Automorph-permutable subgroup]]		* [[Stronger than::Automorph-permutable subgroup]]
	* [[Stronger than::Conjugate-permutable subgroup]]		* [[Stronger than::Conjugate-permutable subgroup]]
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	* [[Stronger than::Elliptic subgroup]]		* [[Stronger than::Elliptic subgroup]]

	=== Relation with normality ===		===Conjunction with other properties===

			In certain kinds of groups:

			* [[Permutable subgroup of finite group]] is a permutable subgroup inside a [[finite group]].

			===Relation with normality===

	Every normal subgroup is permutable, but the converse is not true. In fact, there are groups in which ''every'' subgroup is permutable, but where every subgroup is not normal. These are called [[Quasi-Hamiltonian group\|quasi-Hamiltonian groups]]. In fact, any extension of a cyclic group of prime power order by another cyclic group of prime power order is quasi-Hamiltonian.		Every normal subgroup is permutable, but the converse is not true. In fact, there are groups in which ''every'' subgroup is permutable, but where every subgroup is not normal. These are called [[Quasi-Hamiltonian group\|quasi-Hamiltonian groups]]. In fact, any extension of a cyclic group of prime power order by another cyclic group of prime power order is quasi-Hamiltonian.

Vipul at 16:04, 25 February 2009

2009-02-25T16:04:54Z

← Older revision		Revision as of 16:04, 25 February 2009
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	{{inverse image condition}}		{{inverse image condition}}

	If <math>f:G \to H</math> is a homomorphism and <~~span class="texhtml"~~>''P''</~~span~~> is a permutable subgroup of <~~span class="texhtml"~~>''H''</~~span~~>, then <~~span class="texhtml"~~>''f~~''<sup>~~− 1~~</sup>~~(''P'')</~~span~~> is a permutable subgroup of <~~span class="texhtml"~~>''G''</~~span~~>.		If <math>f:G \to H</math> is a homomorphism and <math>P</math> is a permutable subgroup of <math>H</math>, then <math>f^{− 1}(P)</math> is a permutable subgroup of <math>G</math>

	{{imagecondn}}		{{imagecondn}}

	If <math>f:G \to H</math> is a surjective homomorphism and <~~span class="texhtml"~~>''K''</~~span~~> is a permutable subgroup of <~~span class="texhtml"~~>''G''</~~span~~>, then <~~span class="texhtml"~~>''f''(''K'')</~~span~~> is a permutable subgroup of <~~span class="texhtml"~~>''H''</~~span~~>.		If <math>f:G \to H</math> is a surjective homomorphism and <math>K</math> is a permutable subgroup of <math>G</math>, then <math>f(K)</math> is a permutable subgroup of <math>H</math>. {{proofat\|[[Permutability satisfies image condition]]}}

			{{not upper join-closed}}

			It is possible to have a group <math>G</math>, a subgroup <math>H</math> of <math>G</math>, and intermediate subgroups <math>K_1</math> and <math>K_2</math> such that <math>H</math> is permutable in both <math>K_1</math> and <math>K_2</math> but <math>H</math> is ''not'' permutable in <math>\langle K_1, K_2 \rangle</math>. {{proofat\|[[Permutability is not upper join-closed]]}}
	==Testing==		==Testing==

Vipul: /* Study of the notion */

2008-11-22T22:48:32Z

Study of the notion

← Older revision		Revision as of 22:48, 22 November 2008
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	If <math>f:G \to H</math> is a surjective homomorphism and <span class="texhtml">''K''</span> is a permutable subgroup of <span class="texhtml">''G''</span>, then <span class="texhtml">''f''(''K'')</span> is a permutable subgroup of <span class="texhtml">''H''</span>.		If <math>f:G \to H</math> is a surjective homomorphism and <span class="texhtml">''K''</span> is a permutable subgroup of <span class="texhtml">''G''</span>, then <span class="texhtml">''f''(''K'')</span> is a permutable subgroup of <span class="texhtml">''H''</span>.

			==Testing==

			{{GAP code for subgroup property\|
			test = IsPermutable}}

			Although there's no in-built GAP command for testing permutability, a short snippet of code (available at [[GAP:IsPermutable]]) can be used to create such a function. This is invoked as follows:

			<pre>IsPermutable(group,subgroup);</pre>
	==Study of the notion==		==Study of the notion==

Vipul: /* Relation with other properties */

2008-10-31T22:48:54Z

Relation with other properties

← Older revision		Revision as of 22:48, 31 October 2008
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	* [[Stronger than::Conjugate-permutable subgroup]]		* [[Stronger than::Conjugate-permutable subgroup]]
	* [[Stronger than::Sylow-permutable subgroup]]		* [[Stronger than::Sylow-permutable subgroup]]
			* [[Stronger than::Subnormal-permutable subgroup]]
	* [[Stronger than::Modular subgroup]]: {{proofofstrictimplicationat\|[[Permutable implies modular]]\|[[Modular not implies permutable]]}}		* [[Stronger than::Modular subgroup]]: {{proofofstrictimplicationat\|[[Permutable implies modular]]\|[[Modular not implies permutable]]}}
	* [[Stronger than::Elliptic subgroup]]		* [[Stronger than::Elliptic subgroup]]

Vipul: /* Weaker properties */

2008-09-02T17:26:14Z

Weaker properties

← Older revision		Revision as of 17:26, 2 September 2008
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	* [[Stronger than::Conjugate-permutable subgroup]]		* [[Stronger than::Conjugate-permutable subgroup]]
	* [[Stronger than::Sylow-permutable subgroup]]		* [[Stronger than::Sylow-permutable subgroup]]
	* [[Stronger than::Modular subgroup]]		* [[Stronger than::Modular subgroup]]: {{proofofstrictimplicationat\|[[Permutable implies modular]]\|[[Modular not implies permutable]]}}
	* [[Stronger than::Elliptic subgroup]]		* [[Stronger than::Elliptic subgroup]]

Vipul: /* Metaproperties */

2008-06-12T12:55:22Z

Metaproperties

← Older revision		Revision as of 12:55, 12 June 2008
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	The subgroup generated by a family of permutable subgroups is permutable.		The subgroup generated by a family of permutable subgroups is permutable.

			{{transfercondn}}

			If <math>H</math> is permutable in <math>G</math>, then for any subgroup <math>K \le G</math>, <math>H \cap K</math> is permutable in <math>K</math>.

	{{inverse image condition}}		{{inverse image condition}}

Vipul at 12:47, 12 June 2008

2008-06-12T12:47:14Z

← Older revision		Revision as of 12:47, 12 June 2008
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	# Its [[Defining ingredient::product of subgroups\|product]] with every subgroup of the group is a subgroup		# Its [[Defining ingredient::product of subgroups\|product]] with every subgroup of the group is a subgroup
	# It [[defining ingredient::permuting subgroups\|permutes]] (or commutes) with every subgroup.		# It [[defining ingredient::permuting subgroups\|permutes]] (or commutes) with every subgroup.
			# It permutes with every cyclic subgroup.
	===Definition with symbols===		===Definition with symbols===

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	# For any subgroup <math>K</math> of <math>G</math>, <math>HK</math> (the [[product of subgroups]] <math>H</math> and <math>K</math>) is a group		# For any subgroup <math>K</math> of <math>G</math>, <math>HK</math> (the [[product of subgroups]] <math>H</math> and <math>K</math>) is a group
	# For any subgroup <math>K</math> of <math>G</math>, <math>HK=KH</math>, i.e., <math>H</math> and <math>K</math> are [[permuting subgroups]].		# For any subgroup <math>K</math> of <math>G</math>, <math>HK=KH</math>, i.e., <math>H</math> and <math>K</math> are [[permuting subgroups]].
			# For every <math>g \in G</math>, <math>H</math> permutes with the cyclic subgroup generated by <math>g</math>. In symbols, for every <math>h \in H</math> and <math>g \in G</math>, there exists <math>h' \in H</math> and an integer <math>n</math> such that <math>hg = g^nh'</math>.

	==Formalisms==		==Formalisms==