= Subgroups of polycyclic groups are polycyclic
Given a cyclic-factor <subnormal series> $1=G_0\trianglelefteq\cdots\trianglelefteq G_n=G$ and $H\leq G$, intersect each term with $H$. The induced map $(H\cap G_{i+1})/(H\cap G_i)\to G_{i+1}/G_i$ is <injective>. Since every subgroup of a <cyclic group> is cyclic, the intersection chain is again a cyclic-factor <subnormal series>, after removing repetitions. Hence $H$ is a <polycyclic group>. In particular, every subgroup is finitely generated.
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