Simple group - Revision history

R-a-jones: /* Definition */

2023-12-15T21:12:57Z

Definition

← Older revision		Revision as of 21:12, 15 December 2023
Line 10:		Line 10:
	\| 2 \|\| Surjective homomorphism-based definition \|\| it is nontrivial and any [[Defining ingredient::surjective homomorphism]] from it is either trivial or an isomorphism. \|\| <math>G</math> is nontrivial and any surjective homomorphism <math>\varphi:G \to K</math> is either trivial or an isomorphism.		\| 2 \|\| Surjective homomorphism-based definition \|\| it is nontrivial and any [[Defining ingredient::surjective homomorphism]] from it is either trivial or an isomorphism. \|\| <math>G</math> is nontrivial and any surjective homomorphism <math>\varphi:G \to K</math> is either trivial or an isomorphism.
	\|-		\|-
	\| 3 \|\| Homomorphism-based definition \|\| it is nontrivial and any [[Defining ingredient::homomorphism of groups]] from it is either [[Defining ingredient::trivial homomorphism\|trivial]] or [[injective homomorphism\|injective]] \|\| <math>G</math> is nontrivial and given any homomorphism <math>\varphi:G \to K</math>, <math>\varphi</math> is either injective (i.e., its [[Defining ingredient::kernel]] is trivial) or trivial (i.e., it maps everything to the identity element).		\| 3 \|\| Homomorphism-based definition \|\| it is nontrivial and any [[Defining ingredient::homomorphism of groups]] from it is either [[Defining ingredient::trivial homomorphism of groups\|trivial]] or [[injective homomorphism\|injective]] \|\| <math>G</math> is nontrivial and given any homomorphism <math>\varphi:G \to K</math>, <math>\varphi</math> is either injective (i.e., its [[Defining ingredient::kernel]] is trivial) or trivial (i.e., it maps everything to the identity element).
	\|-		\|-
	\| 4 \|\| Subnormal subgroup-based definition \|\| it is nontrivial and has no proper nontrivial [[defining ingredient::subnormal subgroup]] \|\| <math>G</math> is nontrivial and for any subnormal subgroup <math>H</math> of <math>G</math>, either <math>H</math> is trivial or <math>H = G</math>.		\| 4 \|\| Subnormal subgroup-based definition \|\| it is nontrivial and has no proper nontrivial [[defining ingredient::subnormal subgroup]] \|\| <math>G</math> is nontrivial and for any subnormal subgroup <math>H</math> of <math>G</math>, either <math>H</math> is trivial or <math>H = G</math>.

Vipul: /* Subgroup-defining functions collapse to trivial or improper subgroup */

2014-04-23T20:01:29Z

Subgroup-defining functions collapse to trivial or improper subgroup

← Older revision		Revision as of 20:01, 23 April 2014
Line 102:		Line 102:
	In other words, every nontrivial subgroup of a simple group is [[contranormal subgroup\|contranormal]].		In other words, every nontrivial subgroup of a simple group is [[contranormal subgroup\|contranormal]].

	===Subgroup-defining functions collapse to trivial or ~~improper subgroup~~===		===Subgroup-defining functions collapse to trivial subgroup or whole group===

	Any [[subgroup-defining function]] (such as the [[center]], the [[derived subgroup]], the [[Frattini subgroup]]) returns a [[characteristic subgroup]] of the whole group. In other words, the center, derived subgroup, Frattini subgroup etc. are all characteristic subgroups.		Any [[subgroup-defining function]] (such as the [[center]], the [[derived subgroup]], the [[Frattini subgroup]]) returns a [[characteristic subgroup]] of the whole group. In other words, the center, derived subgroup, Frattini subgroup etc. are all characteristic subgroups.
Line 109:		Line 109:

	* The [[center]] of any simple group is either trivial or the whole group. Hence, every simple group is either [[centerless group\|centerless]] or [[Abelian group\|Abelian]].		* The [[center]] of any simple group is either trivial or the whole group. Hence, every simple group is either [[centerless group\|centerless]] or [[Abelian group\|Abelian]].
	* The [[~~commutator~~ subgroup]] of any simple group is either trivial or the whole group. Hence, every simple group is either [[Abelian group\|Abelian]] or [[perfect group\|perfect]].		* The [[derived subgroup]] of any simple group is either trivial or the whole group. Hence, every simple group is either [[Abelian group\|Abelian]] or [[perfect group\|perfect]].

	===The only simple Abelian groups are cyclic groups of prime order===		===The only simple Abelian groups are cyclic groups of prime order===

Vipul: /* Subgroup-defining functions collapse to trivial or improper subgroup */

2014-04-23T20:00:56Z

Subgroup-defining functions collapse to trivial or improper subgroup

← Older revision		Revision as of 20:00, 23 April 2014
Line 104:		Line 104:
	===Subgroup-defining functions collapse to trivial or improper subgroup===		===Subgroup-defining functions collapse to trivial or improper subgroup===

	Any [[subgroup-defining function]] (such as the [[center]], the [[~~commutator~~ subgroup]], the [[Frattini subgroup]]) returns a [[characteristic subgroup]] of the whole group. In other words, the center, ~~commutator~~ subgroup, Frattini subgroup etc. are all characteristic subgroups.		Any [[subgroup-defining function]] (such as the [[center]], the [[derived subgroup]], the [[Frattini subgroup]]) returns a [[characteristic subgroup]] of the whole group. In other words, the center, derived subgroup, Frattini subgroup etc. are all characteristic subgroups.

	Since [[characteristic implies normal\|every characteristic subgroup is normal]], each of these is also a normal subgroup. But when the whole group is simple, this forces each of these to be either the trivial subgroup or the whole group. Thus, for instance:		Since [[characteristic implies normal\|every characteristic subgroup is normal]], each of these is also a normal subgroup. But when the whole group is simple, this forces each of these to be either the trivial subgroup or the whole group. Thus, for instance:

Vipul: /* Stronger properties */

2014-04-23T19:59:14Z

Stronger properties

← Older revision		Revision as of 19:59, 23 April 2014
Line 45:		Line 45:
	! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Weaker than::Absolutely simple group]] \|\| no proper nontrivial [[serial subgroup]] \|\| \|\| [[simple not implies absolutely simple]] \|\| {{intermediate notions short\|simple group\|absolutely simple group}}		\| [[Weaker than::Absolutely simple group]] \|\| nontrivial, no proper nontrivial [[serial subgroup]] \|\| \|\| [[simple not implies absolutely simple]] \|\| {{intermediate notions short\|simple group\|absolutely simple group}}
	\|-		\|-
	\| [[Weaker than::Strictly simple group]] \|\| no proper nontrivial [[ascendant subgroup]] \|\| \|\| [[simple not implies strictly simple]] \|\| {{intermediate notions short\|simple group\|strictly simple group}}		\| [[Weaker than::Strictly simple group]] \|\| nontrivial, no proper nontrivial [[ascendant subgroup]] \|\| \|\| [[simple not implies strictly simple]] \|\| {{intermediate notions short\|simple group\|strictly simple group}}
	\|-		\|-
	\| [[Weaker than::Simple non-abelian group]] \|\| simple and a [[non-abelian group]] \|\| \|\| \|\| {{intermediate notions short\|simple group\|simple non-abelian group}}		\| [[Weaker than::Simple non-abelian group]] \|\| simple and a [[non-abelian group]] \|\| \|\| \|\| {{intermediate notions short\|simple group\|simple non-abelian group}}

Vipul: /* Examples */

2014-04-23T19:58:42Z

Examples

← Older revision		Revision as of 19:58, 23 April 2014
Line 29:		Line 29:

	* The easiest examples of simple groups are the simple abelian groups. An abelian group is simple if and only if it is [[group of prime order\|cyclic of prime order]].		* The easiest examples of simple groups are the simple abelian groups. An abelian group is simple if and only if it is [[group of prime order\|cyclic of prime order]].
	* The smallest non-abelian simple group is [[alternating group:A5\|the alternating group on five letters]]. This is a group of order 60.		* The smallest non-abelian simple group is [[alternating group:A5\|the alternating group on five letters]]. This is a group of order 60. {{further\|[[A5 is simple]]}}
	* More generally, all [[alternating group]]s on five or ~~more letters~~ are simple. There are other infinite families of simple groups, primarily occurring as linear groups over fields.		* More generally, all [[alternating group]]s of degree five or higher are simple. There are other infinite families of simple groups, primarily occurring as linear groups over fields. {{further\|[[Alternating groups are simple]]}}
	* The finite simple non-abelian groups come in 18 infinite families, and 26 exceptions, termed the [[sporadic simple group]]s. {{further\|[[Classification of finite simple groups]]}}		* The finite simple non-abelian groups come in 18 infinite families, and 26 exceptions, termed the [[sporadic simple group]]s. {{further\|[[Classification of finite simple groups]]}}

Vipul at 01:46, 1 November 2013

2013-11-01T01:46:33Z

← Older revision		Revision as of 01:46, 1 November 2013
Line 8:		Line 8:
	\| 1 \|\| Normal subgroup-based definition \|\| it is nontrivial and has no [[proper subgroup\|proper]] [[nontrivial subgroup\|nontrivial]] [[defining ingredient::normal subgroup]] \|\| <math>G</math> is nontrivial and for any [[normal subgroup]] <math>H</math> of <math>G</math>, either <math>H</math> is trivial or <math>H = G</math>.		\| 1 \|\| Normal subgroup-based definition \|\| it is nontrivial and has no [[proper subgroup\|proper]] [[nontrivial subgroup\|nontrivial]] [[defining ingredient::normal subgroup]] \|\| <math>G</math> is nontrivial and for any [[normal subgroup]] <math>H</math> of <math>G</math>, either <math>H</math> is trivial or <math>H = G</math>.
	\|-		\|-
	\| 2 \|\| Surjective homomorphism-based definition \|\| it is nontrivial and any [[Defining ingredient::surjective homomorphism]] from it is either trivial or an isomorphism. \|\| <math>G</math> is nontrivial and any surjective homomorphism <math>\varphi:G \to K</math> is either trivial, or an isomorphism.		\| 2 \|\| Surjective homomorphism-based definition \|\| it is nontrivial and any [[Defining ingredient::surjective homomorphism]] from it is either trivial or an isomorphism. \|\| <math>G</math> is nontrivial and any surjective homomorphism <math>\varphi:G \to K</math> is either trivial or an isomorphism.
	\|-		\|-
	\| 3 \|\| Homomorphism-based definition \|\| it is nontrivial and any [[Defining ingredient::homomorphism of groups]] from it is either [[Defining ingredient::trivial homomorphism\|trivial]] or [[injective homomorphism\|injective]] \|\| <math>G</math> is nontrivial and given any homomorphism <math>\varphi:G \to K</math>, <math>\varphi</math> is either injective (i.e., its [[Defining ingredient::kernel]] is trivial) or trivial (i.e., it maps everything to the identity element).		\| 3 \|\| Homomorphism-based definition \|\| it is nontrivial and any [[Defining ingredient::homomorphism of groups]] from it is either [[Defining ingredient::trivial homomorphism\|trivial]] or [[injective homomorphism\|injective]] \|\| <math>G</math> is nontrivial and given any homomorphism <math>\varphi:G \to K</math>, <math>\varphi</math> is either injective (i.e., its [[Defining ingredient::kernel]] is trivial) or trivial (i.e., it maps everything to the identity element).

Vipul at 03:09, 17 December 2011

2011-12-17T03:09:21Z

← Older revision		Revision as of 03:09, 17 December 2011
Line 1:		Line 1:
			[[importance rank::2\| ]]
	==Definition==		==Definition==
	{{quick phrase\|prime number among groups, group without any proper nontrivial normal subgroup, group without any proper nontrivial quotients}}		{{quick phrase\|prime number among groups, group without any proper nontrivial normal subgroup, group without any proper nontrivial quotients}}

Vipul at 17:58, 29 June 2010

2010-06-29T17:58:36Z

@@ Line 1: / Line 1: @@
 ==Definition==
 {{quick phrase|prime number among groups, group without any proper nontrivial normal subgroup, group without any proper nontrivial quotients}}
-A nontrivial group is said to be '''simple''' if the following equivalent conditions hold:
+{| class="sortable" border="1"
-# It has no [[proper subgroup|proper]] [[nontrivial subgroup|nontrivial]] [[Defining ingredient::normal subgroup]]
+{{semibasicdef}}
-A nontrivial group <math>G</math> is termed '''simple''' if the following equivalent conditions hold:
+{{pivotal group property}}
-# For any [[normal subgroup]] <math>H</math> of <math>G</math>, <math>H</math> is either trivial or <math>H = G</math>
+{{cfsg}}
 ==Formalisms==

Vipul: /* Relation with other properties */

2010-05-19T23:11:30Z

Relation with other properties

← Older revision		Revision as of 23:11, 19 May 2010
Line 44:		Line 44:
	{{pivotalproperty}}		{{pivotalproperty}}

	* [[:~~Category~~: Variations of ~~simplicity]]~~		* {{#ask: [[Variation of::Simple group]]\|limit = 0\|searchlabel = Variations of simple group}}
	* [[:~~Category~~: ~~Opposites of simplicity~~]]		* {{#ask: [[Analogue of::Simple group]]\|limit = 0\|searchlabel = Analogues of simple group in other algebraic structures}}

	===Stronger properties===		===Stronger properties===

	* [[Weaker than::Absolutely simple group]]		{\| class="sortable" border="1"
	* [[Weaker than::Strictly simple group]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* [[Weaker than::Simple non-abelian group]]		\|-
	* [[Weaker than::Finite simple group]]		\| [[Weaker than::Absolutely simple group]] \|\| no proper nontrivial [[serial subgroup]] \|\| \|\| [[simple not implies absolutely simple]] \|\| {{intermediate notions short\|simple group\|absolutely simple group}}
	* [[Weaker than::Finite simple non-abelian group]]		\|-
			\| [[Weaker than::Strictly simple group]] \|\| no proper nontrivial [[ascendant subgroup]] \|\| \|\| [[simple not implies strictly simple]] \|\| {{intermediate notions short\|simple group\|strictly simple group}}
			\|-
			\| [[Weaker than::Simple non-abelian group]] \|\| simple and a [[non-abelian group]] \|\| \|\| \|\| {{intermediate notions short\|simple group\|simple non-abelian group}}
			\|-
			\| [[Weaker than::Finite simple group]] \|\| simple and a [[finite group]] \|\| \|\| \|\| {{intermediate notions short\|simple group\|finite simple group}}
			\|-
			\| [[Weaker than::Finite simple non-abelian group]] \|\| finite, simple, and non-abelian \|\| \|\| \|\| {{intermediate notions short\|simple group\|finite simple non-abelian group}}
			\|}

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

	* [[Stronger than::Characteristically simple group]]		{\| class="sortable" border="1"
	* [[Stronger than::Directly indecomposable group]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* [[Stronger than::Centrally indecomposable group]]		\|-
	* [[Stronger than::~~Semidirectly indecomposable~~ group]]		\| [[Stronger than::Characteristically simple group]] \|\| nontrivial, no proper nontrivial [[characteristic subgroup]] \|\| \|\| \|\| {{intermediate notions short\|characteristically simple group\|simple group}}
	* [[Stronger than::Primitive group]]		\|-
	* [[Stronger than::Monolithic group]]		\| [[Stronger than::Directly indecomposable group]] \|\| nontrivial, not expressible as an [[internal direct product]] of two proper subgroups \|\| \|\| \|\| {{intermediate notions short\|directly indecomposable group\|simple group}}
	* [[Stronger than::One-headed group]]		\|-
	* [[~~Stronger than::Homosimple group~~]]		\| [[Stronger than::Centrally indecomposable group]] \|\| nontrivial, not expressible as a [[central product]] of two proper subgroups. In other words, every [[central factor]] is either the whole group or is contained in the [[center]]. \|\| \|\| \|\| {{intermediate notions short\|centrally indecomposable group\|simple group}}
	* [[Stronger than::Group of finite composition length]]		\|-
	* [[Stronger than::Normal-comparable group]]		\| [[Stronger than::Splitting-simple group]] \|\| nontrivial, has no proper nontrivial [[complemented normal subgroup]], i.e., is not expressible as an [[internal semidirect product]] of proper subgroups \|\| \|\| \|\| {{intermediate notions short\|splitting-simple group\|simple group}}
	* [[Stronger than::Group satisfying ascending chain condition on subnormal subgroups]]		\|-
			\| [[Stronger than::Primitive group]] \|\| has a [[maximal subgroup]] that is also a [[core-free subgroup]] \|\| [[simple implies primitive]] \|\| [[primitive not implies simple]] \|\| {{intermediate notions short\|primitive group\|simple group}}
			\|-
			\| [[Stronger than::Monolithic group]] \|\| has a unique [[minimal normal subgroup]] contained in every nontrivial normal subgroup \|\| [[simple implies monolithic]] \|\| [[monolithic not implies simple]] \|\| {{intermediate notions short\|monolithic group\|simple group}}
			\|-
			\| [[Stronger than::One-headed group]] \|\| has a unique [[maximal normal subgroup]] that contains every proper normal subgroup \|\| [[simple implies one-headed]] \|\| [[one-headed not implies simple]] \|\| {{intermediate notions short\|one-headed group\|simple group}}
			\|-
			\| [[Stronger than::Group of finite composition length]] \|\| \|\| \|\| \|\| {{intermediate notions short\|group of finite composition length\|simple group}}
			\|-
			\| [[Stronger than::Normal-comparable group]] \|\| the lattice of normal subgroups is totally ordered \|\| \|\| \|\| {{intermediate notions short\|normal-comparable group\|simple group}}
			\|-
			\| [[Stronger than::Group satisfying ascending chain condition on subnormal subgroups]] \|\| any ascending chain of [[subnormal subgroup]]s stabilizes after a finite length \|\| \|\| \|\| {{intermediate notions short\|group satisfying ascending chain condition on subnormal subgroups\|simple group}}
			\|-
			\| [[Stronger than::Group satisfying ascending chain condition on normal subgroups]] \|\| any ascending chain of [[normal subgroup]]s stabilizes after a finite length \|\| \|\| \|\| {{intermediate notions short\|group satisfying ascending chain condition on normal subgroups\|simple group}}
			\|-
			\| [[Stronger than::Group in which every endomorphism is trivial or injective]] \|\| every [[endomorphism]] is either the trivial map or an [[injective endomorphism]] \|\| \|\| \|\| {{intermediate notions short\|group in which every endomorphism is trivial or injective\|simple group}}
			\|-
			\| [[Stronger than::Hopfian group]] \|\| every [[surjective endomorphism]] is an [[automorphism]] \|\| \|\| \|\| {{intermediate notions short\|Hopfian group\|simple group}}
			\|}

	==Facts==		==Facts==

Vipul: /* Stronger properties */

2009-08-21T20:27:01Z

Stronger properties

← Older revision		Revision as of 20:27, 21 August 2009
Line 51:		Line 51:
	* [[Weaker than::Absolutely simple group]]		* [[Weaker than::Absolutely simple group]]
	* [[Weaker than::Strictly simple group]]		* [[Weaker than::Strictly simple group]]
	* [[Weaker than::Simple non-~~Abelian~~ group]]		* [[Weaker than::Simple non-abelian group]]
	* [[Weaker than::Finite simple group]]		* [[Weaker than::Finite simple group]]
	* [[Weaker than::Finite simple non-~~Abelian~~ group]]		* [[Weaker than::Finite simple non-abelian group]]

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

← Older revision		Revision as of 19:58, 23 April 2014
Line 29:		Line 29:

	* The easiest examples of simple groups are the simple abelian groups. An abelian group is simple if and only if it is [[group of prime order\|cyclic of prime order]].		* The easiest examples of simple groups are the simple abelian groups. An abelian group is simple if and only if it is [[group of prime order\|cyclic of prime order]].
	* The smallest non-abelian simple group is [[alternating group:A5\|the alternating group on five letters]]. This is a group of order 60.		* The smallest non-abelian simple group is [[alternating group:A5\|the alternating group on five letters]]. This is a group of order 60. {{further\|[[A5 is simple]]}}
	* More generally, all [[alternating group]]s on five or ~~more letters~~ are simple. There are other infinite families of simple groups, primarily occurring as linear groups over fields.		* More generally, all [[alternating group]]s of degree five or higher are simple. There are other infinite families of simple groups, primarily occurring as linear groups over fields. {{further\|[[Alternating groups are simple]]}}
	* The finite simple non-abelian groups come in 18 infinite families, and 26 exceptions, termed the [[sporadic simple group]]s. {{further\|[[Classification of finite simple groups]]}}		* The finite simple non-abelian groups come in 18 infinite families, and 26 exceptions, termed the [[sporadic simple group]]s. {{further\|[[Classification of finite simple groups]]}}