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 subnormal series with cyclic factors, that is a finite set of subgroups, let's say G₁, ..., G_n+1 such that

G₁ coincides with G

G_n+1 is the trivial subgroup

G_i+1 is a normal subgroup of G_i (for every i between 1 and n)

and the quotient group G_i / G_i+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 ≤ 2, or in other words an extension of a cyclic group by a cyclic group.

Also, polycyclic groups are finitely presented, 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.

A virtually polycyclic group is a group that has a polycyclic subgroup of finite index. Such a group necessarily has a normal polycyclic subgroup of finite index, and therefore such groups are also called polycyclic-by-finite groups. Although polycyclic-by-finite groups need not be solvable, they still have many of the finiteness properties of polycyclic groups; for example, they satisfy the maximal condition, and they are finitely presented and residually finite.

[edit] Polycyclic-by-finite groups

In the textbook (Scott 1964, Ch 7.1) and some papers, an M-group refers to what is now called a polycyclic-by-finite group, which by Hirsch's theorem can also be expressed as a group which has a finite length normal series with each factor a finite group or an infinite cyclic group.

These groups are particularly interesting because they are the only known examples of noetherian group rings (Ivanov 1989), or group rings of finite injective dimension.^{[citation needed]}