Group of nilpotency class two: Difference between revisions

Latest revision as of 02:41, 1 November 2013

Definition

QUICK PHRASES: class two, inner automorphism group is abelian, commutator subgroup inside center, derived subgroup inside center, commutators are central, triple commutators are trivial

Symbol-free definition

A group is said to be of nilpotency class two or nilpotence class two if it satisfies the following equivalent conditions:

Its nilpotency class is at most two, i.e., it is nilpotent of class at most two.
Its derived subgroup (i.e. commutator subgroup) is contained in its center.
The commutator of any two elements of the group is central.
Any triple commutator (i.e., a commutator where one of the terms is itself a commutator) gives the identity element.
Its inner automorphism group is abelian.

NOTE: nilpotency class two is occasionally used to refer to a group whose nilpotency class is precisely two, i.e., a non-abelian group whose nilpotency class is two. This is a more restrictive use of the term than the typical usage, which includes abelian groups.

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 a complete list of semi-basic definitions on this wiki

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

Examples

VIEW: groups satisfying this property | groups dissatisfying this property
VIEW: Related group property satisfactions | Related group property dissatisfactions

Extreme examples

The trivial group is a group of nilpotency class two (in fact, it has class zero).
Any abelian group is a group of nilpotency class two (in fact, it has class one).

Finite examples

By the equivalence of definitions of finite nilpotent group, every finite nilpotent group is a direct product of its Sylow subgroups. Further, if the whole group has class two, so do each of its Sylow subgroups. Thus, every finite group of nilpotency class two is obtained by taking direct products of finite groups of prime power order and class two. So, it suffices to study groups of prime power order and class two. Some salient non-abelian examples are:

For the prime $p=2$ , dihedral group:D8 and quaternion group are two non-abelian groups of class two and order $2^{3}=8$ .
For odd primes $p$ , prime-cube order group:U(3,p) and semidirect product of cyclic group of prime-square order and cyclic group of prime order are (up to isomorphism) the two non-abelian groups of order $p^{3}$ and class two.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Cyclic group	generated by one element			\|FULL LIST, MORE INFO
Abelian group	any two elements commute; center is whole group; derived subgroup is trivial; inner automorphism group is trivial			\|FULL LIST, MORE INFO
Group whose automorphism group is abelian	automorphism group is abelian	abelian automorphism group implies class two	follows from class two not implies abelian automorphism group	\|FULL LIST, MORE INFO
Group whose inner automorphism group is central in automorphism group	inner automorphism group is in center of automorphism group			\|FULL LIST, MORE INFO
Extraspecial group				\|FULL LIST, MORE INFO
Special group				\|FULL LIST, MORE INFO
Frattini-in-center group	commutator subgroup is contained in Frattini subgroup, which is contained in the center			\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
nilpotent group				\|FULL LIST, MORE INFO
metabelian group	abelian normal subgroup with abelian quotient			\|FULL LIST, MORE INFO

Other related properties

Stem group is a group whose center is contained inside its derived subgroup, i.e., the definition of stem group stipulates the reverse containment to that used for the definition of group of nilpotency class two.

Facts

If $G$ is nilpotent of class two, then for any $x\in G$ , the map $y\mapsto [x,y]$ (or alternatively, the map $y\mapsto [y,x]$ ) is an endomorphism of $G$ . Specifically, it is an endomorphism whose image lies inside $G'$ , and we can in fact view the commutator as a biadditive map of Abelian groups:

$G/Z(G)\times G/Z(G)\to G'$

Further information: Class two implies commutator map is endomorphism

@@ Line 1: / Line 1: @@
-{{semibasicdef}}
+[[importance rank::2| ]]
-{{group property}}
 ==Definition==
-{{quick phrase|[[quick phrase::class two]], [[quick phrase::inner automorphism group is abelian]], [[quick phrase::commutator subgroup inside center]], [[quick phrase::commutators are central]], [[quick phrase::triple commutators are trivial]]}}
+{{quick phrase|[[quick phrase::class two]], [[quick phrase::inner automorphism group is abelian]], [[quick phrase::commutator subgroup inside center]], [[quick phrase::derived subgroup inside center]], [[quick phrase::commutators are central]], [[quick phrase::triple commutators are trivial]]}}
 ===Symbol-free definition===
@@ Line 9: / Line 8: @@
 # Its [[defining ingredient::nilpotency class]] is at most two, i.e., it is nilpotent of class at most two.
-# Its [[defining ingredient::commutator subgroup]] (i.e. derived subgroup) is contained in its [[defining ingredient::center]]
+# Its [[defining ingredient::derived subgroup]] (i.e. commutator subgroup) is contained in its [[defining ingredient::center]].
-# The [[commutator]] of any two elements of the group is central
+# The [[commutator]] of any two elements of the group is central.
-# Any triple commutator (i.e., a commutator where one of the terms is itself a commutator) gives the identity element
+# Any triple commutator (i.e., a commutator where one of the terms is itself a commutator) gives the identity element.
 # Its [[defining ingredient::inner automorphism group]] is [[defining ingredient::abelian group|abelian]].
-Note that sometimes, ''nilpotency class two'' is used to refer to a group whose nilpotency class is precisely two, i.e., a non-abelian group whose nilpotency class is two.
+{{quotation|'''NOTE''': ''nilpotency class two'' is occasionally used to refer to a group whose nilpotency class is precisely two, i.e., a non-abelian group whose nilpotency class is two. This is a more restrictive use of the term than the typical usage, which includes abelian groups.}}
+{{semibasicdef}}
+{{group property}}
+==Examples==
+{{group property see examples}}
+===Extreme examples===
+* The [[trivial group]] is a group of nilpotency class two (in fact, it has class zero).
+* Any [[abelian group]] is a group of nilpotency class two (in fact, it has class one).
+===Finite examples===
+By the [[equivalence of definitions of finite nilpotent group]], every finite nilpotent group is a direct product of its [[Sylow subgroup]]s. Further, if the whole group has class two, so do each of its Sylow subgroups. Thus, every finite group of nilpotency class two is obtained by taking direct products of finite groups of prime power order and class two. So, it suffices to study groups of prime power order and class two. Some salient ''non-abelian'' examples are:
+* For the prime <math>p = 2</math>, [[dihedral group:D8]] and [[quaternion group]] are two non-abelian groups of class two and order <math>2^3 = 8</math>.
+* For odd primes <math>p</math>, [[prime-cube order group:U(3,p)]] and [[semidirect product of cyclic group of prime-square order and cyclic group of prime order]] are (up to isomorphism) the two non-abelian groups of order <math>p^3</math> and class two.
 ==Relation with other properties==
@@ Line 25: / Line 43: @@
 | [[Weaker than::Cyclic group]] || generated by one element || || || {{intermediate notions short|group of nilpotency class two|cyclic group}}
 |-
-| [[Weaker than::Abelian group]] || any two elements commute; center is whole group; commutator subgroup is trivial; inner automorphism group is trivial || || || {{intermediate notions short|group of nilpotency class two|abelian group}}
+| [[Weaker than::Abelian group]] || any two elements commute; center is whole group; derived subgroup is trivial; inner automorphism group is trivial || || || {{intermediate notions short|group of nilpotency class two|abelian group}}
 |-
-| [[Weaker than::Aut-abelian group]] || automorphism group is abelian || [[aut-abelian implies class two]] || [[class two not implies aut-abelian]] || {{intermediate notions short|group of nilpotency class two|aut-abelian group}}
+| [[Weaker than::Group whose automorphism group is abelian]] || automorphism group is abelian || [[abelian automorphism group implies class two]] || follows from [[class two not implies abelian automorphism group]] || {{intermediate notions short|group of nilpotency class two|group whose automorphism group is abelian}}
 |-
 | [[Weaker than::Group whose inner automorphism group is central in automorphism group]] || inner automorphism group is in center of automorphism group || || || {{intermediate notions short|group of nilpotency class two|group whose inner automorphism group is central in automorphism group}}
@@ Line 43: / Line 61: @@
 ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 |-
-| [[Stronger than::Nilpotent group]] || || || || {{intermediate notions short|nilpotent group|group of nilpotency class two}}
+| [[Stronger than::nilpotent group]] || || || || {{intermediate notions short|nilpotent group|group of nilpotency class two}}
 |-
-| [[Stronger than::Metabelian group]] || abelian normal subgroup with abelian quotient || || || {{intermediate notions short|metabelian group|group of nilpotency class two}}
+| [[Stronger than::metabelian group]] || abelian normal subgroup with abelian quotient || || || {{intermediate notions short|metabelian group|group of nilpotency class two}}
 |}
+===Other related properties===
+* [[Stem group]] is a group whose [[center]] is contained inside its [[derived subgroup]], i.e., the definition of stem group stipulates the reverse containment to that used for the definition of group of nilpotency class two.
 ==Facts==