PlanetMath: polycyclic group

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polycyclic group

(Definition)

A group is said to be polycyclic if it has a subnormal series

$\displaystyle \{1\}=G_0\triangleleft G_1\triangleleft \dots\triangleleft G_{n-1}\triangleleft G_n=G$

such that $G_{i+1}/G_i$ is cyclic for each $i=0,\dots,n-1$ . (Note that this differs from the definition of a supersolvable group in that it does not require each

to be normal in

.) A subnormal series of this form is called a polycyclic series.

Polycyclic groups are obviously solvable. In fact, the polycyclic groups are precisely those solvable groups that satisfy the maximal condition (that is, those solvable groups all of whose subgroups are finitely generated). In particular, a finite group is polycyclic if and only if it is solvable.

The Hirsch length (or Hirsch number, named after Kurt Hirsch) of a polycyclic group is the number of infinite factors in a polycyclic series of . This is independent of the choice of polycyclic series, as a consequence of the Schreier Refinement Theorem. More generally, the Hirsch length of a polycyclic-by-finite group is the Hirsch length of a polycyclic normal subgroup of finite index in (all such subgroups having the same Hirsch length). J. A. Hillman[1] has further extended the concept of Hirsch length to cover all elementary amenable groups.

References

1: Jonathan A. Hillman, Elementary amenable groups and 4-manifolds with Euler characteristic 0, J. Austral. Math. Soc. (Series A) 50 (1991), 160-170. (This paper can be viewed on the Australian Mathematical Society website.)

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