# 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.

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 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