Simple group
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
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This article defines a group property that is pivotal (i.e., important) among existing group properties
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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
- 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. Further information: A5 is simple
- More generally, all alternating groups of degree five or higher are simple. There are other infinite families of simple groups, primarily occurring as linear groups over fields. Further information: Alternating groups are simple
- 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
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 | Locally finite simple group|FULL LIST, MORE INFO | ||
Finite simple non-abelian group | finite, simple, and non-abelian | |FULL LIST, MORE INFO |
Weaker properties
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 center of any simple group is either trivial or the whole group. Hence, every simple group is either centerless or Abelian.
- The derived 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)
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