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'polycyclic group'
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| Title of object: |
polycyclic group |
| Canonical Name: |
PolycyclicGroup |
| Type: |
Definition |
| Created on: |
2004-10-03 09:34:54 |
| Modified on: |
2004-10-03 09:46:22 |
| Classification: |
msc:20F16 |
| Defines: |
polycyclic, polycyclic series, Hirsch number |
Revision comment (for changes between this and next version):
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\def\normal{\triangleleft} |
Content:
\PMlinkescapeword{factors}
\PMlinkescapeword{independent}
\PMlinkescapeword{refinement}
\PMlinkescapeword{satisfy}
\PMlinkescapeword{theorem}
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 number} 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. |
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