quasicyclic group
QuasicyclicGroup
2005-11-25 15:07:55
2006-03-23 15:12:32
Definition
quasicyclic
quasi-cyclic
Pr\"ufer p-group
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Let $p$ be a prime number.
The \emph{$p$-quasicyclic group} (or \emph{Pr\"ufer $p$-group}, or \emph{$p^\infty$ group}) is the $p$-primary component of $\Q/\Z$,
that is, the unique maximal \PMlinkname{$p$-subgroup}{PGroup4} of $\Q/\Z$.
Any \PMlinkname{group}{Group} isomorphic to this will also be called a $p$-quasicyclic group.
The $p$-quasicyclic group will be denoted by $\Z(p^\infty)$.
Other notations in use include $\Z[p^\infty]$, $\Z/p^\infty\Z$,
$\Z_{p^\infty}$ and $C_{p^\infty}$.
$\Z(p^\infty)$ may also be defined in a number of other (equivalent) ways
(again, up to isomorphism):
\begin{itemize}
\item $\Z(p^\infty)$ is
the group of all $p^n$-th complex roots of $1$, for $n\in\N$.
\item $\Z(p^\infty)$ is
the injective hull of $\Z/p\Z$ (viewing abelian groups as $\Z$-\PMlinkname{modules}{Module}).
\item $\Z(p^\infty)$ is the direct limit of the groups $\Z/p^n\Z$.
\end{itemize}
A \emph{quasicyclic group} (or \emph{Pr\"ufer group}) is
a group that is $p$-quasicyclic for some prime $p$.
The \PMlinkname{subgroup}{Subgroup} structure of $\Z(p^\infty)$ is particularly simple:
all proper subgroups are finite and cyclic,
and there is exactly one of order $p^n$ for each non-negative integer $n$.
In particular,
this means that the subgroups are linearly ordered by inclusion,
and all subgroups are fully invariant.
The quasicyclic groups are
the only infinite groups with a linearly ordered subgroup lattice.
They are also
the only infinite solvable groups whose proper subgroups are all finite.
Quasicyclic groups are locally cyclic, \PMlinkname{divisible}{DivisibleGroup} and co-Hopfian.
Every infinite locally cyclic $p$-group is isomorphic to $\Z(p^\infty)$.