# polycyclic group

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

$$\{1\}={G}_{0}\u25c1{G}_{1}\u25c1\mathrm{\dots}\u25c1{G}_{n-1}\u25c1{G}_{n}=G$$ |

such that ${G}_{i+1}/{G}_{i}$ is cyclic for each $i=0,\mathrm{\dots},n-1$.
(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 |
---|---|

Canonical name | PolycyclicGroup |

Date of creation | 2013-03-22 14:40:50 |

Last modified on | 2013-03-22 14:40:50 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 14 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 20F16 |

Related topic | SupersolvableGroup |

Defines | polycyclic |

Defines | polycyclic series |

Defines | Hirsch number |

Defines | Hirsch length |