Let \(G\) be a group with identity \(e\). A central series of \(G\) is a finite chain of subgroups \[ G = G^0 \geq G^1 \geq \cdots \geq G^n = \theset{e} \] with each \(G^i \normal G\) and \([G, G^i] \leq G^{i+1}\) for \(0\leq i < n\), where \([G, G^i]\) is the subgroup generated by the commutators \(ghg^{-1}h^{-1}\) with \(g\in G\) and \(h\in G^i\).
For subgroups \(G^{i+1}\leq G^i\) normal in \(G\), the condition \([G, G^i] \leq G^{i+1}\) holds if and only if \(G^i/G^{i+1}\) is contained in the center of \(G/G^{i+1}\).