Simple group

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
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This article defines a group property that is pivotal (i.e., important) among existing group properties
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VIEW RELATED: Group property implications | Group property non-implications | Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions |Group property dissatisfactions<random> @@@
RANDOM GROUP PROPERTY: <random>Simple group: A nontrivial group having no proper nontrivial normal subgroup.@@@Noetherian group: A group where every subgroup is finitely generated.@@@Complete group: A group for which the natural homomorphism to its automorphism group is an isomorphism, i.e., a centerless group for which every automorphism is inner.@@@SQ-universal group: A group for which every finitely generated group can be realized as a subquotient.@@@T-group: A group where any subnormal subgroup is normal, i.e., where normality is transitive. In general, normality is not transitive.@@@Locally finite group: A group in which every finitely generated subgroup is finite.@@@Group satisfying normalizer condition: A group with no proper self-normalizing subgroup.</random></random>

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

History

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Definition

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

Symbol-free definition

A nontrivial group is said to be simple if the following equivalent conditions hold:

1. It has no proper nontrivial normal subgroup
2. Any surjective homomorphism from it is either trivial, or an isomorphism.
3. Any homomorphism from it is either trivial or injective
4. It has no proper nontrivial subnormal subgroup

Definition with symbols

A nontrivial group $G$ is termed simple if the following equivalent conditions hold:

1. For any normal subgroup $H$ of $G$, $H$ is either trivial or $H=G$
2. Any surjective homomorphism $\varphi :G\to K$ is either trivial, or an isomorphism.
3. Given any homomorphism $\varphi :G\to K$, $\varphi$ is either injective (i.e., its kernel is trivial) or trivial (i.e., it maps everything to the identity element).
4. For any subnormal subgroup $H$ of $G$, $H$ is either trivial or $H=G$

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

• The easiest examples of simple groups are the simple abelian groups. An abelian group is simple if and only if it is cyclic of prime order.
• The smallest non-abelian simple group is the alternating group on five letters. This is a group of order 60.
• More generally, all alternating groups on five or more letters are simple. There are other infinite families of simple groups, primarily occurring as linear groups over fields.
• The finite simple non-abelian groups come in 18 infinite families, and 26 exceptions, termed the sporadic simple groups. Further information: Classification of finite simple groups

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

• Variations of simple group
• Analogues of simple group in other algebraic structures

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Absolutely simple group	no proper nontrivial serial subgroup		simple not implies absolutely simple	|FULL LIST, MORE INFO
Strictly simple group	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 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.

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 center of any simple group is either trivial or the whole group. Hence, every simple group is either centerless or Abelian.
• The commutator subgroup of any simple group is either trivial or the whole group. Hence, every simple group is either Abelian or perfect.

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

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Definition links

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