Group of nilpotency class two

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 defining ingredient::nilpotency class is at most two, i.e., it is nilpotent of class at most two.
 * 2) Its defining ingredient::derived subgroup (i.e. commutator subgroup) is contained in its defining ingredient::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 defining ingredient::inner automorphism group is abelian.

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.

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:

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