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Contents 1 Definition 1.1 Symbol-free definition 2 Examples 2.1 Extreme examples 2.2 Finite examples 3 Relation with other properties 3.1 Stronger properties 3.2 Weaker properties 4 Facts

This article is about a standard (though not very rudimentary) definition in group theory.[SHOW MORE]

The article text may, however, contain more than just the basic definition
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This article defines a group property: a property that can be evaluated to true/false for any given 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

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:

1. Its nilpotency class is at most two, i.e., it is nilpotent of class at most two.
2. Its derived subgroup (i.e. commutator subgroup) is contained in its center
3. The commutator of any two elements of the group is central
4. Any triple commutator (i.e., a commutator where one of the terms is itself a commutator) gives the identity element
5. 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.

Examples

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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 2³ = 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 p³ 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			click here
Abelian group	any two elements commute; center is whole group; commutator subgroup is trivial; inner automorphism group is trivial			click here
Aut-abelian group	automorphism group is abelian	aut-abelian implies class two	class two not implies aut-abelian	click here
Group whose inner automorphism group is central in automorphism group	inner automorphism group is in center of automorphism group
Extraspecial group				click here
Special group				click here
Frattini-in-center group	commutator subgroup is contained in Frattini subgroup, which is contained in the center

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Nilpotent group
Metabelian group	abelian normal subgroup with abelian quotient

Facts

If G is nilpotent of class two, then for any , the map (or alternatively, the map ) 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:

Further information: Class two implies commutator map is endomorphism