Group of nilpotency class two
QUICK PHRASES: class two, inner automorphism group is abelian, commutator subgroup inside center, derived subgroup inside center, commutators are central, triple commutators are trivial
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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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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- 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).
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 , dihedral group:D8 and quaternion group are two non-abelian groups of class two and order .
- For odd primes , 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 and class two.
Relation with other properties
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|nilpotent group||Group of nilpotency class three|FULL LIST, MORE INFO|
|metabelian group||abelian normal subgroup with abelian quotient|||FULL LIST, MORE INFO|
- 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.
If is nilpotent of class two, then for any , the map (or alternatively, the map ) is an endomorphism of . Specifically, it is an endomorphism whose image lies inside , and we can in fact view the commutator as a biadditive map of Abelian groups:
Further information: Class two implies commutator map is endomorphism