Perfect group: Difference between revisions

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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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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>

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 $g_{1}, g_{2}, \dots, g_{2 n - 1}, g_{2 n}$ (with possible repetitions) such that $g = [g_{1}, g_{2}] [g_{3}, g_{4}] \dots [g_{2 n - 1}, g_{2 n}]$ .
3	trivial abelianization	its abelianization is a trivial group.	the abelianization $G^{a b} = G / [G, G] = H_{1} (G; Z)$ 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 $φ : G \to K$ , $φ$ 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.

Nomenclature disambiguation

Leinster groups are sometimes called perfect groups in some literature, unrelated to this notion.

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:

Here are some even more complicated/less basic groups satisfying the property:

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:

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 $⟨ H_{i} ⟩_{i \in I}$ 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	\|FULL LIST, MORE INFO
quasisimple group	perfect and inner automorphism group is a simple non-abelian 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	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, $\overset{x}{E}$ 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 $r^{t h}$ 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 978-0471433347^{More info}	174		definition introduced in exercise (Problem 19)
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613^{More 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 0521786754^{More info}	27			Google Books

External links

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Definition links

@@ Line 1: / Line 1: @@
-{{group property}}
 {{semibasicdef}}
+{{pivotal group property}}
 ==Definition==
-===Symbol-free definition===
+A [[group]] is said to be '''perfect''' if it satisfies the following equivalent conditions:
+{| class="sortable" border="1"
+! No. !! Shorthand !! A group is said to be perfect if ... !! A group <math>G</math> is said to be perfect if ...
+|-
+| 1 || equals own derived subgroup || it equals its own [[defining ingredient::derived subgroup]] (i.e., its commutator subgroup with itself). || <math>G</math> equals the derived subgroup <math>[G,G]</math>, sometimes also denoted <math>G'</math>.
+|-
+| 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 <math>g \in G</math>, there exist elements <math>g_1,g_2,\dots,g_{2n-1},g_{2n}</math> (with possible repetitions) such that <math>g = [g_1,g_2][g_3,g_4] \dots [g_{2n-1},g_{2n}]</math>.
+|-
+| 3 || trivial abelianization || its [[defining ingredient::abelianization]] is a [[trivial group]]. || the abelianization <math>G^{\operatorname{ab}} = G/[G,G] = H_1(G;\mathbb{Z})</math> 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 <math>K</math>, and any homomorphism of groups <math>\varphi:G \to K</math>, <math>\varphi</math> must send all elements of <math>G</math> to the identity element of <math>K</math>.
+|}
+{{tabular definition 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]].
+==Nomenclature disambiguation==
+[[Leinster group]]s are sometimes called perfect groups in some literature, unrelated to this notion.
+==Examples==
+===Extreme examples===
+* The trivial group is a [[perfect group]].
+===Groups satisfying the property===
+{{groups satisfying property sorted by importance rank}}
+===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.
-A [[group]] is said to be '''perfect''' if it equals its own [[commutator subgroup]].
+{{groups dissatisfying property sorted by importance rank}}
-===Definition with symbols===
+==Metaproperties==
-A [[group]] <math>G</math> is said to be '''perfect''' if <math>[G,G] = G</math>.
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
+|-
+| [[dissatisfies metaproperty::subgroup-closed group property]] || No || [[perfectness is not subgroup-closed]] || It is possible to have a perfect group <math>G</math> and a subgroup <math>H</math> of <math>G</math> such that <math>H</math> 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]]).
+|-
+| [[dissatisfies metaproperty::characteristic subgroup-closed group property]] || No || [[perfectness is not characteristic subgroup-closed]] || It is possible to have a perfect group <math>G</math> and a [[characteristic subgroup]] <math>H</math> of <math>G</math> such that <math>H</math> is not perfect.
+|-
+| [[satisfies metaproperty::quotient-closed group property]] || Yes || [[perfectness is quotient-closed]] || If <math>G</math> is a perfect group and <math>H</math> is a [[normal subgroup]] of <math>G</math>, the [[quotient group]] <math>G/H</math> is also a perfect group.
+|-
+| [[satisfies metaproperty::finite direct product-closed group property]] || Yes || [[perfectness is finite direct product-closed]] || Suppose <math>G_1,G_2,\dots,G_n</math> are all (possibly isomorphic, possibly non-isomorphic) perfect groups. Then, the [[external direct product]] <math>G_1 \times G_2 \times \dots \times G_n</math> is also a perfect group.
+|-
+| [[satisfies metaproperty::join-closed group property]] || Yes || [[perfectness is join-closed]] || Suppose <math>H_i, i \in I</math> is a (possibly finite, possibly infinite) collection of subgroups of a group <math>G</math>, such that each <math>H_i</math> is a perfect group. Then, the [[join of subgroups]] <math>\langle H_i \rangle_{i \in I}</math> is a perfect group.
+|}
-===In terms of the fixed-point operator===
+==Relation with other properties==
+===Stronger properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::superperfect group]] || perfect and [[Schur-trivial group|Schur-trivial]]: the [[Schur multiplier]] is trivial. || (by definition) || follows from [[perfect not implies Schur-trivial]] || {{intermediate notions short|perfect group|superperfect group}}
+|-
+| [[Weaker than::simple non-abelian group]] || non-abelian and has no proper nontrivial normal subgroup. || || [[simple and non-abelian implies perfect]] || {{intermediate notions short|perfect group|simple non-abelian group}}
+|-
+| [[Weaker than::quasisimple group]] || perfect and [[inner automorphism group]] is a [[simple non-abelian group]] || || || {{intermediate notions short|perfect group|quasisimple group}}
+|-
+| [[Weaker than::group in which every element is a commutator]] || every element of the group is a [[commutator]] of two elements of the group. || || || {{intermediate notions short|perfect group|group in which every element is a commutator}}
+|}
+===Weaker properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::square-in-derived group]] || every [[square element]] is in the [[derived subgroup]]. ||  || || {{intermediate notions short|square-in-derived group|perfect group}}
+|-
+| [[Stronger than::stem group]] || the [[center]] is contained in the [[derived subgroup]] || || || {{intermediate notions short|stem group|perfect group}}
+|-
+| [[Stronger than::group with unique Schur covering group]] || the [[Schur covering group]] is unique, or equivalently, <math>\operatorname{Ext}^1</math> of the abelianization over the Schur multiplier is trivial. || || || {{intermediate notions short|group with unique Schur covering group|perfect group}}
+|}
+==Testing==
+{{GAP command for gp|
+test = IsPerfectGroup|
+listbyorder = PerfectGroup}}
-The property of being perfect is obtained by applying the [[fixed-point operator]] to a [[subgroup-defining function]], namely the [[commutator subgroup]].
+To test whether a given group is perfect, the command is:
-==Property theory==
+<pre>IsPerfectGroup(group);</pre>
-{{DP-closed}}
+The command:
-A direct product of perfect groups is a perfect group. This follows from the fact that the commutator subgroup of a direct product is the direct product of the commutator subgroups.
+<pre>PerfectGroup(n,r)</pre>
-{{Q-closed}}
+gives the <math>r^{th}</math> perfect group of order <math>n</math>. If <math>r</math> is not specified, this simply gives the first perfect group of order <math>n</math>. An error is thrown if there are no perfect groups of order <math>n</math>.
-Any quotient of a perfect group is perfect. Thus, the property of being perfect is a [[quotient-hereditary subgroup property]].
+==References==
+===Textbook references===
-===Subgroup-generation===
+{| class="sortable" border="1"
+! Book !! Page number !! Chapter and section !! Contextual information !! View
+|-
+| {{booklink-defined-tabular|DummitFoote|174||definition introduced in exercise}} (Problem 19) ||
+|-
+| {{booklink-defined-tabular|RobinsonGT|157|Section 5.4, Problem 4|definition introduced in exercise}} || [http://books.google.com/books?id=Zl3mGxwUtmEC&pg=PA157 Google Books]
+|-
+| {{booklink-defined-tabular|FGTAsch|27}} || [http://books.google.com/books?id=BprbtnlI6HEC&pg=PA27 Google Books]
+|}
-The subgroup generated by any two perfect subgroups of a group is also a perfect subgroup.
+==External links==
+{{searchbox|"perfect+group"+derived+OR+commutator}}
+===Definition links===
+* {{wp|Perfect group}}
+* {{mathworld|PerfectGroup}}
+* {{planetmath|PerfectGroup}}