Polycyclic group

A group is polycyclic when it has a subnormal series with cyclic factors. Every subgroup and quotient of a polycyclic group is finitely generated, and extensions of polycyclic groups are polycyclic.

If a group has a cyclic-factor subnormal series of length $n$, choose one lift of a generator of each nontrivial factor. Induction gives $G_i=⟨G_{i-1},g_i⟩$, 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.

Given a cyclic-factor subnormal series $1=G_0⊴\cdots ⊴G_n=G$ and $H\le G$, 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.

Every polycyclic group has a finite-index subgroup with a series whose nontrivial factors are infinite cyclic. This is the polycyclic finite-index lemma; it also supplies torsion-free finite-index subgroups. Such a subgroup is a poly-infinite cyclic group. Combining its series with the extension rule for Poincare duality groups makes it a duality group of dimension its Hirsch length.

The Hirsch length of a polycyclic group counts infinite cyclic factors in a polycyclic series. Refinement shows it is independent of the series, additive in extensions, and unchanged in finite-index subgroups. A finite-index poly-infinite cyclic group has one infinite cyclic factor for each unit of Hirsch length.

A poly-infinite cyclic group has a finite subnormal series all of whose nontrivial factors are infinite cyclic groups. Intersecting its series with a subgroup again gives trivial or infinite cyclic factors; repetitions may be deleted. Splicing a kernel series with the preimage of a quotient series proves closure under extensions. Such groups are finitely presented groups: induct on the series, split the last extension by an infinite cyclic quotient, and use finite presentation of a semidirect product.