Simple group: Difference between revisions

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{{semibasicdef}}
[[importance rank::2| ]]
 
{{group property}}
 
{{cfsg}}
 
==History==
 
{{history missing}}
 
==Definition==
==Definition==
{{quick phrase|prime number among groups, group without any proper nontrivial normal subgroup, group without any proper nontrivial quotients}}


===Symbol-free definition===
{| class="sortable" border="1"
! No. !! Shorthand !! A group is simple if ... !! A group <math>G</math> is simple if ...
|-
| 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.
|-
| 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>.
|}


A nontrivial group is said to be '''simple''' if the following equivalent conditions hold:
{{semibasicdef}}


* It has no [[proper subgroup|proper]] [[nontrivial subgroup|nontrivial]] [[normal subgroup]]
{{pivotal group property}}
* Any homomorphism from it is either [[trivial homomorphism|trivial]] or [[injective homomorphism|injective]]


===Definition with symbols===
{{cfsg}}


A nontrivial group <math>G</math> is termed '''simple''' if the following equivalent conditions hold:
==Formalisms==
{{obtainedbyapplyingthe|simple group operator|normal subgroup}}


* For any normal subgroup <math>H</math> of <math>G</math>, <math>H</math> is either trivial or the whole group.
The [[group property]] of being simple is obtained by applying the [[simple group operator]] to the [[subgroup property]] of [[normality]].
* Given any homomorphism <math>\phi:G \to H</math>, <math>\phi</math> is either injective (that is, its kernel is trivial) or trivial (that is, it maps everything to the identity element).


{{obtainedbyapplyingthe|simple group operator|normal subgroup}}
==Examples==


The [[group property]] of being simple is obtained by applying the [[simple group operator]] to the [[subgroup property]] of [[normality]].
* 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. {{further|[[A5 is simple]]}}
* 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]]}}


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


* [[Absolutely simple group]]
{| class="sortable" border="1"
* [[Strictly simple group]]
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
* [[Simple non-Abelian group]]
|-
* [[Finite 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}}
* [[Finite simple non-Abelian 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::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===


* [[Characteristically simple group]]
{| class="sortable" border="1"
* [[Directly indecomposable group]]
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
* [[Centrally indecomposable group]]
|-
* [[Semidirectly indecomposable group]]
| [[Stronger than::Characteristically simple group]] || nontrivial, no proper nontrivial [[characteristic subgroup]] || || || {{intermediate notions short|characteristically simple group|simple group}}
* [[Primitive group]]
|-
* [[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}}
* [[One-headed group]]
|-
* [[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::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::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==
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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 [[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:


* 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===
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{{further|[[Simplicity testing problem]]}}
{{further|[[Simplicity testing problem]]}}


{{GAP command for gp|IsSimple}}
{{GAP command for gp|
test = IsSimpleGroup}}
 
To determine on GAP whether a given group is simple:


To determine on GAP whether a give group is simple:
<pre>IsSimpleGroup (group)</pre>


<pre>IsSimple (group)</pre>
where <tt>group</tt> could either be a definition of a group or a name for a group already defined.


where <pre>group</pre> could either be a definition of a group or a name for a group already defined.
==References==
===Textbook references===
* {{booklink-defined|DummitFoote}}, Page 91
* {{booklink-defined|RobinsonGT}}, Page 16
* {{booklink-defined|Artin}}, Page 201, between points (2.3) and (2.4) (definition introduced in paragraph)


==External links==
==External links==
{{searchbox|"simple+group"}}
{{searchbox|"simple+group"}}
===Definition links===
* {{wp|Simple group}}
* {{mathworld|SimpleGroup}}
* {{planetmath|Simple4}}

Latest revision as of 21:12, 15 December 2023

Definition

QUICK PHRASES: prime number among groups, group without any proper nontrivial normal subgroup, group without any proper nontrivial quotients

No. Shorthand A group is simple if ... A group is simple if ...
1 Normal subgroup-based definition it is nontrivial and has no proper nontrivial normal subgroup is nontrivial and for any normal subgroup of , either is trivial or .
2 Surjective homomorphism-based definition it is nontrivial and any surjective homomorphism from it is either trivial or an isomorphism. is nontrivial and any surjective homomorphism is either trivial or an isomorphism.
3 Homomorphism-based definition it is nontrivial and any homomorphism of groups from it is either trivial or injective is nontrivial and given any homomorphism , is either injective (i.e., its 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 subnormal subgroup is nontrivial and for any subnormal subgroup of , either is trivial or .


This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
VIEW: Definitions built on this | Facts about this: (facts closely related to Simple group, all facts related to Simple group) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a complete list of semi-basic definitions on this wiki

This article defines a group property that is pivotal (i.e., important) among existing group properties
View a list of pivotal group properties | View a complete list of group properties [SHOW MORE]

This article is about a term related to the Classification of finite simple groups

Formalisms

In terms of the simple group operator

This property is obtained by applying the simple group operator to the property: normal subgroup
View other properties obtained by applying the simple group operator

The group property of being simple is obtained by applying the simple group operator to the subgroup property of normality.

Examples

Relation with other properties

This property is a pivotal (important) member of its property space. Its variations, opposites, and other properties related to it and defined using it are often studied

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Absolutely simple group nontrivial, no proper nontrivial serial subgroup simple not implies absolutely simple |FULL LIST, MORE INFO
Strictly simple group nontrivial, no proper nontrivial ascendant subgroup simple not implies strictly simple |FULL LIST, MORE INFO
Simple non-abelian group simple and a non-abelian group |FULL LIST, MORE INFO
Finite simple group simple and a finite group |FULL LIST, MORE INFO
Finite simple non-abelian group finite, simple, and non-abelian |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Characteristically simple group nontrivial, no proper nontrivial characteristic subgroup |FULL LIST, MORE INFO
Directly indecomposable group nontrivial, not expressible as an internal direct product of two proper subgroups |FULL LIST, MORE INFO
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. |FULL LIST, MORE INFO
Splitting-simple group nontrivial, has no proper nontrivial complemented normal subgroup, i.e., is not expressible as an internal semidirect product of proper subgroups |FULL LIST, MORE INFO
Primitive group has a maximal subgroup that is also a core-free subgroup simple implies primitive primitive not implies simple |FULL LIST, MORE INFO
Monolithic group has a unique minimal normal subgroup contained in every nontrivial normal subgroup simple implies monolithic monolithic not implies simple |FULL LIST, MORE INFO
One-headed group has a unique maximal normal subgroup that contains every proper normal subgroup simple implies one-headed one-headed not implies simple |FULL LIST, MORE INFO
Group of finite composition length |FULL LIST, MORE INFO
Normal-comparable group the lattice of normal subgroups is totally ordered |FULL LIST, MORE INFO
Group satisfying ascending chain condition on subnormal subgroups any ascending chain of subnormal subgroups stabilizes after a finite length |FULL LIST, MORE INFO
Group satisfying ascending chain condition on normal subgroups any ascending chain of normal subgroups stabilizes after a finite length |FULL LIST, MORE INFO
Group in which every endomorphism is trivial or injective every endomorphism is either the trivial map or an injective endomorphism |FULL LIST, MORE INFO
Hopfian group every surjective endomorphism is an automorphism |FULL LIST, MORE INFO

Facts

Proper subgroups are core-free

In a simple group, the normal core of any subgroup is a normal subgroup, and hence is either the whole group or the trivial subgroup. Thus, the normal core of any proper subgroup must be the trivial subgroup.

In other words, every proper subgroup is core-free.

Nontrivial subgroups are contranormal

In a simple group, the normal closure of any subgroup is either the whole group or the trivial subgroup. Thus, the normal closure of any nontrivial subgroup is the whole group.

In other words, every nontrivial subgroup of a simple group is contranormal.

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.

Since 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:

The only simple Abelian groups are cyclic groups of prime order

The proof of this follows more or less directly from the fact that in a simple Abelian group, every subgroup is normal, and hence, the subgroup generated by any nonidentity element is normal. This forces that the whole group is cyclic generated by any element, and hence it must be cyclic of prime order.

Metaproperties

Direct products

A direct product of simple groups is not simple. In fact, the two direct factors are themselves normal subgroups.

Subgroups

Every finite group occurs as a subgroup of some simple group. Hence the property of being embeddable as a subgroup of a simple group is nothing distinguishing.

Quotients

The only quotients of a simple group are itself and the trivial group.

Testing

The testing problem

Further information: Simplicity testing problem

GAP command

This group property can be tested using built-in functionality of Groups, Algorithms, Programming (GAP).
The GAP command for this group property is:IsSimpleGroup
View GAP-testable group properties

To determine on GAP whether a given group is simple: 

IsSimpleGroup (group)

where group could either be a definition of a group or a name for a group already defined.

References

Textbook references

  • Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, More info, Page 91
  • A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, More info, Page 16
  • Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, More info, Page 201, between points (2.3) and (2.4) (definition introduced in paragraph)

External links

Search for "simple+group" on the World Wide Web:
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Definition links

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