Perfect 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
VIEW: Definitions built on this | Facts about this: (facts closely related to Perfect group, all facts related to Perfect 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]
Definition
A group is said to be perfect if it satisfies the following equivalent conditions:
No. | Shorthand | A group is said to be perfect if ... | A group ![]() |
---|---|---|---|
1 | equals own derived subgroup | it equals its own derived subgroup (i.e., its commutator subgroup with itself). | ![]() ![]() ![]() |
2 | every element is a product of commutators | every element of the group can be expressed as a product in the group of finitely many elements each of which is a commutator. | for any ![]() ![]() ![]() |
3 | trivial abelianization | its abelianization is a trivial group. | the abelianization ![]() |
4 | trivial homomorphism to any abelian group | any homomorphism of groups from it to an abelian group is the trivial homomorphism. | for any abelian group ![]() ![]() ![]() ![]() ![]() |
This definition is presented using a tabular format. |View all pages with definitions in tabular format
In terms of the fixed-point operator
The property of being perfect is obtained by applying the fixed-point operator to a subgroup-defining function, namely the derived subgroup.
Examples
Extreme examples
- The trivial group is a perfect group.
Groups satisfying the property
Here are some basic/important groups satisfying the property:
Here are some relatively less basic/important groups satisfying the property:
GAP ID | |
---|---|
Alternating group:A5 | 60 (5) |
Alternating group:A6 | 360 (118) |
Projective special linear group:PSL(3,2) | 168 (42) |
Special linear group:SL(2,5) | 120 (5) |
Here are some even more complicated/less basic groups satisfying the property:
GAP ID | |
---|---|
Alternating group:A7 | |
Projective special linear group:PSL(2,11) | 660 (13) |
Projective special linear group:PSL(2,8) | 504 (156) |
Special linear group:SL(2,7) | 336 (114) |
Special linear group:SL(2,9) | 720 (409) |
Groups dissatisfying the property
Note that any nontrivial solvable group cannot be a perfect group, so this gives lots of non-examples. The examples discussed below concentrate more on the non-solvable groups that still fail to be perfect.
Here are some basic/important groups that do not satisfy the property:
Here are some relatively less basic/important groups that do not satisfy the property:
GAP ID | |
---|---|
Symmetric group:S5 | 120 (34) |
Here are some even more complicated/less basic groups that do not satisfy the property:
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
subgroup-closed group property | No | perfectness is not subgroup-closed | It is possible to have a perfect group ![]() ![]() ![]() ![]() |
characteristic subgroup-closed group property | No | perfectness is not characteristic subgroup-closed | It is possible to have a perfect group ![]() ![]() ![]() ![]() |
quotient-closed group property | Yes | perfectness is quotient-closed | If ![]() ![]() ![]() ![]() |
finite direct product-closed group property | Yes | perfectness is finite direct product-closed | Suppose ![]() ![]() |
join-closed group property | Yes | perfectness is join-closed | Suppose ![]() ![]() ![]() ![]() |
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
superperfect group | perfect and Schur-trivial: the Schur multiplier is trivial. | (by definition) | follows from perfect not implies Schur-trivial | |FULL LIST, MORE INFO |
simple non-abelian group | non-abelian and has no proper nontrivial normal subgroup. | simple and non-abelian implies perfect | Quasisimple group, Semisimple group|FULL LIST, MORE INFO | |
quasisimple group | perfect and inner automorphism group is a simple non-abelian group | Semisimple group|FULL LIST, MORE INFO | ||
group in which every element is a commutator | every element of the group is a commutator of two elements of the group. | |FULL LIST, MORE INFO |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
square-in-derived group | every square element is in the derived subgroup. | |FULL LIST, MORE INFO | ||
stem group | the center is contained in the derived subgroup | |FULL LIST, MORE INFO | ||
group with unique Schur covering group | the Schur covering group is unique, or equivalently, ![]() |
|FULL LIST, MORE INFO |
Testing
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:IsPerfectGroup
View GAP-testable group properties
To test whether a given group is perfect, the command is:
IsPerfectGroup(group);
The command:
PerfectGroup(n,r)
gives the perfect group of order
. If
is not specified, this simply gives the first perfect group of order
. An error is thrown if there are no perfect groups of order
.
References
Textbook references
Book | Page number | Chapter and section | Contextual information | View |
---|---|---|---|---|
Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347More info | 174 | definition introduced in exercise (Problem 19) | ||
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613More info | 157 | Section 5.4, Problem 4 | definition introduced in exercise | Google Books |
Finite Group Theory (Cambridge Studies in Advanced Mathematics) by Michael Aschbacher, ISBN 0521786754More info | 27 | Google Books |
External links
Search for "perfect+group"+derived+OR+commutator on the World Wide Web:
Scholarly articles: Google Scholar, JSTOR
Books: Google Books, Amazon
This wiki: Internal search, Google site search
Encyclopaedias: Wikipedia (or using Google), Citizendium
Math resource pages:Mathworld, Planetmath, Springer Online Reference Works
Math wikis: Topospaces, Diffgeom, Commalg, Noncommutative
Discussion fora: Mathlinks, Google Groups
The web: Google, Yahoo, Windows Live
Learn more about using the Searchbox OR provide your feedback