Since the center $Z(G)$ is a normal subgroup of $G$, we may define by induction (with the help of this proposition here) a series of inclusions of normal subgroups $Z^k(G) \subseteq G$ where $Z^0(G)$ is the trivial subgroup and $Z^k(G)$ is the inverse image of the center $Z(G/Z^{k-1}(G))$ along the canonical homomorphism $G \to G/Z^{k-1}(G)$. The resulting series

is called the upper central series of $G$, and Proposition shows that in the case of a finite $p$-group, this series consists of strict inclusions that eventually terminate in the full subgroup $G$. A group with that property is a nilpotent group. In particular it is a solvable group.