Almost simple group - Revision history

Vipul at 03:10, 17 December 2011

2011-12-17T03:10:02Z

← Older revision		Revision as of 03:10, 17 December 2011
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	~~{{group property}}~~		[[importance rank::3\| ]]

	~~{{variationof~~\|~~simplicity}}~~

	==Definition==		==Definition==

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	* There exists a normal subgroup <math>N</math> of <math>G</math> such that <math>N</math> is a simple non-abelian group and <math>C_G(N)</math> is trivial.		* There exists a normal subgroup <math>N</math> of <math>G</math> such that <math>N</math> is a simple non-abelian group and <math>C_G(N)</math> is trivial.

			{{group property}}
			{{variationof\|simplicity}}
	==Relation with other properties==		==Relation with other properties==

Vipul: /* Definition */

2009-08-27T22:37:56Z

Definition

← Older revision		Revision as of 22:37, 27 August 2009
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	A [[group]] is said to be '''almost simple''' if it satisfies the following equivalent conditions:		A [[group]] is said to be '''almost simple''' if it satisfies the following equivalent conditions:

	* There is a [[simple non-~~Abelian~~ group]] such that the given group can be embedded between the simple group and its automorphism group.		* There is a [[simple non-abelian group]] such that the given group can be embedded between the simple group and its automorphism group.
	* The group has a [[centralizer-free subgroup\|centralizer-free]] non-~~Abelian~~ [[simple normal subgroup]].		* The group has a [[centralizer-free subgroup\|centralizer-free]] non-abelian [[simple normal subgroup]].
	===Definition with symbols===		===Definition with symbols===

	A [[group]] <math>G</math> is said to be '''almost simple''' if it satisfies the following equivalent conditions:		A [[group]] <math>G</math> is said to be '''almost simple''' if it satisfies the following equivalent conditions:

	* There is a [[simple group\|simple]] non-~~Abelian~~ group <math>S</math> such that <math>S \le T \le \operatorname{Aut}(S)</math> for some group <math>T</math> isomorphic to <math>G</math>.		* There is a [[simple group\|simple]] non-abelian group <math>S</math> such that <math>S \le T \le \operatorname{Aut}(S)</math> for some group <math>T</math> isomorphic to <math>G</math>.
	* There exists a normal subgroup <math>N</math> of <math>G</math> such that <math>N</math> is a simple non-~~Abelian~~ group and <math>C_G(N)</math> is trivial.		* There exists a normal subgroup <math>N</math> of <math>G</math> such that <math>N</math> is a simple non-abelian group and <math>C_G(N)</math> is trivial.

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

Vipul at 21:03, 2 January 2009

2009-01-02T21:03:44Z

← Older revision		Revision as of 21:03, 2 January 2009
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	===Symbol-free definition===		===Symbol-free definition===

	A [[group]] is said to be '''almost simple''' if ~~there is a [[simple group\|simple]] non-Abelian group such that~~ the ~~given group can be embedded between the simple group and its automorphism group.~~		A [[group]] is said to be '''almost simple''' if it satisfies the following equivalent conditions:

			* There is a [[simple non-Abelian group]] such that the given group can be embedded between the simple group and its automorphism group.
			* The group has a [[centralizer-free subgroup\|centralizer-free]] non-Abelian [[simple normal subgroup]].
	===Definition with symbols===		===Definition with symbols===

	A [[group]] <math>G</math> is said to be '''almost simple''' if ~~there~~ is a [[simple group\|simple]] non-Abelian group <math>S</math> such that <math>S \le T \le \operatorname{Aut}(S)</math> for some group <math>T</math> isomorphic to <math>G</math>.		A [[group]] <math>G</math> is said to be '''almost simple''' if it satisfies the following equivalent conditions:

			* There is a [[simple group\|simple]] non-Abelian group <math>S</math> such that <math>S \le T \le \operatorname{Aut}(S)</math> for some group <math>T</math> isomorphic to <math>G</math>.
			* There exists a normal subgroup <math>N</math> of <math>G</math> such that <math>N</math> is a simple non-Abelian group and <math>C_G(N)</math> is trivial.

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

Vipul at 19:24, 2 January 2009

2009-01-02T19:24:15Z

← Older revision		Revision as of 19:24, 2 January 2009
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	===Definition with symbols===		===Definition with symbols===

	A [[group]] <math>G</math> is said to be '''almost simple''' if there is a [[simple group\|simple]] non-Abelian group <math>S</math> such that <math>S</math> ~~≤~~ <math>G</math> ~~≤~~ <math>~~Aut(S)~~</math>.		A [[group]] <math>G</math> is said to be '''almost simple''' if there is a [[simple group\|simple]] non-Abelian group <math>S</math> such that <math>S \le T \le \operatorname{Aut}(S)</math> for some group <math>T</math> isomorphic to <math>G</math>.

	==Relation with other properties==		==Relation with other properties==
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	* [[Simple non-Abelian group]]		* [[Simple non-Abelian group]]

			==Facts==

			* [[Automorphism group of simple non-Abelian group is complete]]
			* [[Almost simple not implies simple or complete]]: An almost simple group need not be either simple or complete: in other words, it can be properly sandwiched between a simple group and its automorphism group.
			* [[Symmetric groups are almost simple]]: For <math>n \ge 5</math>, the symmetric group on <math>n</math> letters is almost simple. Note that for <math>n \ne 6</math>, it is in fact the whole automorphism group.

Vipul: 1 revision

2008-05-07T22:49:52Z

1 revision

← Older revision	Revision as of 22:49, 7 May 2008
(No difference)

Vipul at 15:31, 6 March 2007

2007-03-06T15:31:51Z

New page

{{group property}}

{{variationof|simplicity}}

==Definition==

===Symbol-free definition===

A [[group]] is said to be '''almost simple''' if there is a [[simple group|simple]] non-Abelian group such that the given group can be embedded between the simple group and its automorphism group.

===Definition with symbols===

A [[group]] <math>G</math> is said to be '''almost simple''' if there is a [[simple group|simple]] non-Abelian group <math>S</math> such that <math>S</math> ≤ <math>G</math> ≤ <math>Aut(S)</math>.

==Relation with other properties==

===Stronger properties===

* [[Simple non-Abelian group]]