polycyclic groupPolycyclicGroup2004-10-03 09:34:542006-03-24 15:56:19Definitionpolycyclicpolycyclic seriesHirsch numberHirsch length\usepackage{amssymb}
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A group $G$ is said to be \emph{polycyclic} if it has a subnormal series
\[\{1\}=G_0\normal G_1\normal\dots\normal G_{n-1}\normal G_n=G\]
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 \emph{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 \PMlinkname{subgroups}{Subgroup} are finitely generated).
In particular, a finite group is polycyclic if and only if it is solvable.
The \emph{Hirsch length} (or \emph{Hirsch number}, named after \PMlinkexternal{Kurt Hirsch}{http://www-history.mcs.st-and.ac.uk/history//Biographies/Hirsch.html})
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\cite{hillman} has further extended the concept of Hirsch length to cover all elementary amenable groups.
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\bibitem{hillman}
Jonathan A.~Hillman,
\PMlinkescapetext{{\sl 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 \PMlinkexternal{on the Australian Mathematical Society website}{http://anziamj.austms.org.au/JAMSA/V50/Part1/Hillman/p0160.html}.)
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