Is upper central series of a group unique?

Question

Let G be a group. Its upper central series is defined inductively $$Z_0=\{1\}$$ $$Z_{i+1}/Z_i=Z(G/Z_i)$$ Existence of $Z_{i+1}$ is guaranteed by correspondence theorem. But can we say anything about its uniqueness? If the series is not unique then how can we define nilpotence class of nilpotent group based on smallest $c$ for which $Z_c=G$?

Answer 1

I'm not sure what are you talking about when you mention "existence"; but definition definitely gives you unique subgroup series.

Maybe it's better to write down a definition of UCS in a way similar to lower central series to avoid mentioning different quotient groups: $Z_0 = 1$, and $Z_{k+1}$ consists of elements $c$ such that $[g, c] \in Z_k$ for all $g \in G$.

Answer 2

For any group $G$ and every normal subgroup $N\trianglelefteq G$ define the sets $$\mathcal{G} = \{A: N\subseteq A < G\}$$ and $$\mathcal{H} = \{S: S < G/N\}$$ The correspondence theorem tells you that the map $\phi: \mathcal{G} \to \mathcal{H}$ defined by $\phi(A)=A/N$ for all $A\in \mathcal{G}$ is a bijection between the sets $\mathcal{G}$ and $\mathcal{H}$. Therefore, for any subgroup $S$ of the quotient $G/N$ there exists some unique subgroup $A$ of $G$ that contains $N$ and $A/N = S$, namely $A=\phi^{-1}(S)$.

This is enough to prove both the existence and the uniqueness of the subgroups $Z_i$.