Perfect group

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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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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 G is said to be perfect if ...
1 equals own derived subgroup it equals its own derived subgroup (i.e., its commutator subgroup with itself). G equals the derived subgroup [G,G], sometimes also denoted G'.
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 g \in G, there exist elements g_1,g_2,\dots,g_{2n-1},g_{2n} (with possible repetitions) such that g = [g_1,g_2][g_3,g_4] \dots [g_{2n-1},g_{2n}].
3 trivial abelianization its abelianization is a trivial group. the abelianization G^{\operatorname{ab}} = G/[G,G] = H_1(G;\mathbb{Z}) is the trivial group.
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 K, and any homomorphism of groups \varphi:G \to K, \varphi must send all elements of G to the identity element of K.
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

Groups satisfying the property

Here are some basic/important groups satisfying the property:

 GAP ID
Trivial group1 (1)

Here are some relatively less basic/important groups satisfying the property:

 GAP ID
Alternating group:A560 (5)
Alternating group:A6360 (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:S5120 (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 G and a subgroup H of G such that H is not perfect. In fact, any nontrivial perfect group has a nontrivial cyclic subgroup that is not perfect. (in fact, every finite group is a subgroup of a finite perfect group).
characteristic subgroup-closed group property No perfectness is not characteristic subgroup-closed It is possible to have a perfect group G and a characteristic subgroup H of G such that H is not perfect.
quotient-closed group property Yes perfectness is quotient-closed If G is a perfect group and H is a normal subgroup of G, the quotient group G/H is also a perfect group.
finite direct product-closed group property Yes perfectness is finite direct product-closed Suppose G_1,G_2,\dots,G_n are all (possibly isomorphic, possibly non-isomorphic) perfect groups. Then, the external direct product G_1 \times G_2 \times \dots \times G_n is also a perfect group.
join-closed group property Yes perfectness is join-closed Suppose H_i, i \in I is a (possibly finite, possibly infinite) collection of subgroups of a group G, such that each H_i is a perfect group. Then, the join of subgroups \langle H_i \rangle_{i \in I} is a perfect group.

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, \operatorname{Ext}^1 of the abelianization over the Schur multiplier is trivial. |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 r^{th} perfect group of order n. If r is not specified, this simply gives the first perfect group of order n. An error is thrown if there are no perfect groups of order n.

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

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