Permutable complements - Revision history

Mvlancellotti: /* Permutable complements need not be unique */

2016-10-22T20:49:43Z

Permutable complements need not be unique

← Older revision		Revision as of 20:49, 22 October 2016
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	===Permutable complements need not be unique===		===Permutable complements need not be unique===

	Given a subgroup <math>H</math> of <math>G</math>, there may or may not exist permutable complements of <math>H</math>. Moreover, there may exist multiple possibilities for a complement to <math>H</math>, and the multiple possibilities ~~may~~ not ~~even~~ be pairwise isomorphic.		Given a subgroup <math>H</math> of <math>G</math>, there may or may not exist permutable complements of <math>H</math>. Moreover, there may exist multiple possibilities for a complement to <math>H</math>, and the multiple possibilities might not be pairwise isomorphic.

	{{further\|[[Every group of given order is a permutable complement for symmetric groups]], [[Retract not implies normal complements are isomorphic]]}}		{{further\|[[Every group of given order is a permutable complement for symmetric groups]], [[Retract not implies normal complements are isomorphic]]}}

Mvlancellotti: /* Symbol-free definition */

2016-10-22T20:48:15Z

Symbol-free definition

← Older revision		Revision as of 20:48, 22 October 2016
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	===Symbol-free definition===		===Symbol-free definition===

	Two ~~subgroup~~ of a group are said to be permutable complements if:		Two subgroups of a group are said to be permutable complements if:

	* Their intersection is trivial		* Their intersection is trivial

Vipul: /* Other related facts */

2009-04-22T19:04:25Z

Other related facts

← Older revision		Revision as of 19:04, 22 April 2009
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	* [[Retract not implies every permutable complement is normal]]		* [[Retract not implies every permutable complement is normal]]
	* [[Permutable complement to normal subgroup is isomorphic to quotient]]		* [[Permutable complement to normal subgroup is isomorphic to quotient]]
	* [[Permutable complements to abelian normal subgroup are ~~isomorphic~~]]		* [[Permutable complements to abelian normal subgroup are automorphic]]
	* [[Retract not implies normal complements are isomorphic]]		* [[Retract not implies normal complements are isomorphic]]
	* [[Permutable complements to normal subgroup need not be automorphic]]		* [[Permutable complements to normal subgroup need not be automorphic]]

Vipul at 19:02, 22 April 2009

2009-04-22T19:02:33Z

← Older revision		Revision as of 19:02, 22 April 2009
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	* [[There may be multiple subgroups that are pairwise permutable complements]]		* [[There may be multiple subgroups that are pairwise permutable complements]]
	* [[Retract not implies every permutable complement is normal]]		* [[Retract not implies every permutable complement is normal]]
			* [[Permutable complement to normal subgroup is isomorphic to quotient]]
			* [[Permutable complements to abelian normal subgroup are isomorphic]]
			* [[Retract not implies normal complements are isomorphic]]
			* [[Permutable complements to normal subgroup need not be automorphic]]

Vipul: /* Facts */

2008-10-31T22:37:36Z

Facts

← Older revision		Revision as of 22:37, 31 October 2008
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	===Other related facts===		===Other related facts===

	* [[There ~~can~~ be multiple subgroups that are pairwise permutable complements]]		* [[There may be multiple subgroups that are pairwise permutable complements]]
	* [[Retract not implies every permutable complement is normal]]		* [[Retract not implies every permutable complement is normal]]

Vipul: /* Facts */

2008-10-31T22:23:45Z

Facts

← Older revision		Revision as of 22:23, 31 October 2008
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	Given a subgroup <math>H</math> of <math>G</math>, there may or may not exist permutable complements of <math>H</math>. Moreover, there may exist multiple possibilities for a complement to <math>H</math>, and the multiple possibilities may not even be pairwise isomorphic.		Given a subgroup <math>H</math> of <math>G</math>, there may or may not exist permutable complements of <math>H</math>. Moreover, there may exist multiple possibilities for a complement to <math>H</math>, and the multiple possibilities may not even be pairwise isomorphic.

	{{further\|[[Every group of given order is a permutable complement for symmetric groups]], [[Retract not implies ~~permutable~~ complements are isomorphic]]}}		{{further\|[[Every group of given order is a permutable complement for symmetric groups]], [[Retract not implies normal complements are isomorphic]]}}

	===For a normal subgroup, they are fixed upto isomorphism===		===For a normal subgroup, they are fixed upto isomorphism===

Vipul: /* Facts */

2008-10-31T22:12:35Z

Facts

← Older revision		Revision as of 22:12, 31 October 2008
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	Given a subgroup <math>H</math> of <math>G</math>, there may or may not exist permutable complements of <math>H</math>. Moreover, there may exist multiple possibilities for a complement to <math>H</math>, and the multiple possibilities may not even be pairwise isomorphic.		Given a subgroup <math>H</math> of <math>G</math>, there may or may not exist permutable complements of <math>H</math>. Moreover, there may exist multiple possibilities for a complement to <math>H</math>, and the multiple possibilities may not even be pairwise isomorphic.

	{{further\|[[Every group of given order is a permutable complement for symmetric groups]]}}		{{further\|[[Every group of given order is a permutable complement for symmetric groups]], [[Retract not implies permutable complements are isomorphic]]}}

	===For a normal subgroup, they are fixed upto isomorphism===		===For a normal subgroup, they are fixed upto isomorphism===

	Interestingly, when a subgroup is [[normal subgroup\|normal]], then any two permutable complements to it must be isomorphic. In fact, any permutable complement to it must be isomorphic to the [[quotient group]].		Interestingly, when a subgroup is [[normal subgroup\|normal]], then any two permutable complements to it must be isomorphic. In fact, any permutable complement to it must be isomorphic to the [[quotient group]].

			===Other related facts===

			* [[There can be multiple subgroups that are pairwise permutable complements]]
			* [[Retract not implies every permutable complement is normal]]

Vipul: 2 revisions

2008-05-08T00:00:35Z

2 revisions

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

Vipul: /* Permutable complements need not be unique */

2007-05-30T08:58:32Z

Permutable complements need not be unique

← Older revision		Revision as of 08:58, 30 May 2007
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	Given a subgroup <math>H</math> of <math>G</math>, there may or may not exist permutable complements of <math>H</math>. Moreover, there may exist multiple possibilities for a complement to <math>H</math>, and the multiple possibilities may not even be pairwise isomorphic.		Given a subgroup <math>H</math> of <math>G</math>, there may or may not exist permutable complements of <math>H</math>. Moreover, there may exist multiple possibilities for a complement to <math>H</math>, and the multiple possibilities may not even be pairwise isomorphic.

	{{further\|[[Every ~~subgroup~~ of given order is a permutable complement for symmetric groups]]}}		{{further\|[[Every group of given order is a permutable complement for symmetric groups]]}}

	===For a normal subgroup, they are fixed upto isomorphism===		===For a normal subgroup, they are fixed upto isomorphism===

	Interestingly, when a subgroup is [[normal subgroup\|normal]], then any two permutable complements to it must be isomorphic. In fact, any permutable complement to it must be isomorphic to the [[quotient group]].		Interestingly, when a subgroup is [[normal subgroup\|normal]], then any two permutable complements to it must be isomorphic. In fact, any permutable complement to it must be isomorphic to the [[quotient group]].

Vipul at 08:53, 30 May 2007

2007-05-30T08:53:37Z

New page

{{subgroup symmrel}}

==Definition==

===Symbol-free definition===

Two subgroup of a group are said to be permutable complements if:

* Their intersection is trivial
* Their [[product of subgroups|product]] is the whole group

===Definition with symbols===

Two subgroups <math>H</math> and <math>K</math> of a group <math>G</math> are termed '''permutable complements''' if the following two conditions hold:

* <math>H \cap K</math> is the trivial group
* <math>HK = G</math>

==Facts==

===Permutable complements need not be unique===

Given a subgroup <math>H</math> of <math>G</math>, there may or may not exist permutable complements of <math>H</math>. Moreover, there may exist multiple possibilities for a complement to <math>H</math>, and the multiple possibilities may not even be pairwise isomorphic.

{{further|[[Every subgroup of given order is a permutable complement for symmetric groups]]}}

===For a normal subgroup, they are fixed upto isomorphism===

Interestingly, when a subgroup is [[normal subgroup|normal]], then any two permutable complements to it must be isomorphic. In fact, any permutable complement to it must be isomorphic to the [[quotient group]].