= Poly-infinite cyclic group
= Poly-Z group
{synonym}
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>.
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