Perfect group

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RANDOM GROUP PROPERTY: 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.

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

• The trivial group is a perfect group.

Groups satisfying the property

Here are some basic/important groups satisfying the property:

	GAP ID
Trivial group	1 (1)

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 $G$ 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

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Definition links

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