Polycyclic group
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In mathematics and group theory, a polycyclic group is a solvable group that satisfies the maximal condition on subgroups (that is, every subgroup is finitely generated).
Equivalently, a group G is polycyclic if and only if it admits a series with cyclic factors, that is a finite set of subgroups, let's say G1, ..., Gn+1 such that
- G1 coincides with G
- Gn+1 is the trivial subgroup
- Gi+1 is a normal subgroup of Gi (for every i between 1 and n)
- and the quotient group Gi / Gi+1 is a cyclic group (for every i between 1 and n)
Needless to say, this is where their name comes from. A metacyclic group is, according to the current standard definition[1], a polycyclic group with n = 1, or in other words an extension of a cyclic group by a cyclic group.
Also, polycyclic groups are precisely the finitely presented solvable groups, and this makes them very interesting from a computational point of view.
Examples of polycyclic groups include finitely generated abelian groups, finitely generated nilpotent groups, and finite solvable groups.
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