Infinite cyclic group has no chief series

A hypothetical finite chief series in $\mathbb{Z}$ has a first nonzero term $n\mathbb{Z}$. The subgroup $2n\mathbb{Z}$ is normal in $\mathbb{Z}$ and strictly between zero and that term. This contradicts the defining maximality condition for the first chief factor. Finite groups always admit chief series, by choosing a maximal chain of their normal subgroups; this example shows why finiteness matters.