The upper central series starts at the identity and repeatedly lifts the center of the current quotient. It gives the most direct canonical definition of a nilpotent group. [@isaacs2008]
Definition
\(Z_1(G)=Z(G)\). These terms should not be confused with centralizers of elements or subgroups.
Core result
The equivalent definition is the quotient-center condition. Inductively, if \(N_i\leq Z_i\), centrality of the next factor places \(N_{i+1}\) in \(Z_{i+1}\). Comparison with the lower central series gives the nilpotency-class criterion.
Structural properties
- Every \(Z_i(G)\) is characteristic.
- If a finite group has nontrivial center in every nontrivial quotient, the upper central series grows strictly until it reaches \(G\).
- Normalizer growth in a nilpotent group follows by choosing the first \(Z_i(G)\) not contained in a proper subgroup.
- A quotient can have larger upper-central terms than \(Z_i(G)N/N\); equality is not automatic.
- Every upper-central factor is abelian, so nilpotent groups are solvable.
The upper series emphasizes successive central extensions, while the lower series emphasizes vanishing of long commutators. They meet in the nilpotency class.
Example and boundary
Knowledge network
The lower central series gives the dual criterion, the nilpotent-group core article proves finite equivalences, and the center is the first term.
Proof and sources
Definition and finite center criteria are in Theorems 1.20-1.22; comparison with commutator series is in Section 4C. [@isaacs2008]
