# polycyclic group

amenable groups

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

 $\{1\}=G_{0}\triangleleft G_{1}\triangleleft\dots\triangleleft G_{n-1}% \triangleleft G_{n}=G$

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 $G_{i}$ to be normal in $G$.) 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 (http://planetmath.org/Subgroup) 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 http://www-history.mcs.st-and.ac.uk/history//Biographies/Hirsch.htmlKurt Hirsch) of a polycyclic group $G$ is the number of infinite factors in a polycyclic series of $G$. 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 $G$ is the Hirsch length of a polycyclic normal subgroup of finite index in $G$ (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, , J. Austral. Math. Soc. (Series A) 50 (1991), 160–170. (This paper can be viewed http://anziamj.austms.org.au/JAMSA/V50/Part1/Hillman/p0160.htmlon the Australian Mathematical Society website.)
Title polycyclic group PolycyclicGroup 2013-03-22 14:40:50 2013-03-22 14:40:50 yark (2760) yark (2760) 14 yark (2760) Definition msc 20F16 SupersolvableGroup polycyclic polycyclic series Hirsch number Hirsch length