Polycyclic group

Definition
A group is said to be polycyclic if it satisfies the following equivalent conditions:


 * 1) It has a defining ingredient::subnormal series (wherein each member is normal in its successor) of finite length, starting from the trivial group and ending at the whole group, such that all the successive quotients are cyclic.
 * 2) It is a defining ingredient::solvable group and all the factor groups between successive members of its derived series are defining ingredient::finitely generated abelian groups.
 * 3) It is both a defining ingredient::Noetherian group (also called a slender group, i.e., every subgroup is finitely generated) and a defining ingredient::solvable group.

Definition with symbols
A group $$G$$ is said to be polycyclic if there exists a series of subgroups:

$$1 = H_0 \triangleleft H_1 \triangleleft H_2 \triangleleft \ldots \triangleleft H_n = G$$

where each $$H_{i+1}/H_i$$ is cyclic.

Origin of the concept
Hirsch, in the years 1938-1954, obtained results on polycylic groups, but he used the term S-group for them.

Origin of the term
The term polycyclic group was first used by Hall in 1954. It is now the commonly accepted term.

Metaproperties
Any subgroup of a polycyclic group is polycyclic. The subnormal serise for the subgroup is obtained by intersecting with it the subnormal series of the whole group.

Any quotient of a polycyclic group is polycyclic. The subnormal series for the quotient is obtained by taking the image of the subnormal series for the whole group, via the quotient map.

A finite direct product of polycyclic groups is polycyclic.