periodic groupPeriodicGroup2005-12-01 16:52:132007-07-25 10:03:03Definitionperiodictorsion\usepackage{amssymb}
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A group $G$ is said to be \emph{periodic} (or \emph{torsion})
if every element of $G$ is of finite order.
All finite groups are periodic.
More generally, all locally finite groups are periodic.
Examples of periodic groups that are not locally finite include Tarski groups,
and Burnside groups $B(m,n)$ of odd exponent $n\ge665$ on $m>1$ generators.
Some easy results on periodic groups:
\begin{theorem}
\item Every \PMlinkname{subgroup}{Subgroup} of a periodic group is periodic.
\end{theorem}
\begin{theorem}
\item Every \PMlinkname{quotient}{QuotientGroup} of a periodic group is periodic.
\end{theorem}
\begin{theorem}
\item Every \PMlinkname{extension}{GroupExtension} of a periodic group by a periodic group is periodic.
\end{theorem}
\begin{theorem}
\item Every restricted direct product of periodic groups is periodic.
\end{theorem}
Note that (unrestricted) direct products of periodic groups are not necessarily periodic. For example, the direct product of all finite cyclic groups $\Z/n\Z$ is not periodic, as the element that is $1$ in every coordinate has infinite order.
Some further results on periodic groups:
\begin{theorem}
Every solvable periodic group is locally finite.
\end{theorem}
\begin{theorem}
Every periodic abelian group is the direct sum of its maximal \PMlinkname{$p$-groups}{PGroup4} over all primes $p$.
\end{theorem}