Infinite cyclic group has no chief series
= 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.