Upper central series
The upper central series of G is 1 = Z₀ ⊆ Z₁ = Z(G) ⊆ Z₂ ⊆ …, with
Zk+1 / Zk = Z(G / Zk). It ascends from the center; G is
nilpotent exactly when it reaches G, in the same number of
steps as the lower central series.
Its limit Z∞(G) is the hypercenter. For a non-nilpotent G it stops
below G — e.g. Z∞ = 1 whenever Z(G) = 1 (as for S₃ or any
simple non-abelian group).