= Generator bound from a cyclic-factor series
{title2=$d(G)\leq n$}
If a <group> has a cyclic-factor <subnormal series> of length $n$, choose one lift of a generator of each nontrivial factor. Induction gives $G_i=\langle G_{i-1},g_i\rangle$, so the chosen lifts generate $G$. Thus at most $n$ generators are needed. Applying this to the intersection chain in <subgroups of polycyclic groups are polycyclic> bounds the number of generators of every subgroup by the same series length.
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