| Version current |
Version 7 |
| \PMlinkescapeword{coordinate} |
\PMlinkescapeword{coordinate} |
| \PMlinkescapeword{obvious} |
\PMlinkescapeword{obvious} |
| \PMlinkescapeword{subgroup} |
\PMlinkescapeword{subgroup} |
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| A group $G$ is said to be \emph{periodic} (or \emph{torsion}) |
A group $G$ is said to be \emph{periodic} (or \emph{torsion}) |
| if every element of $G$ is of finite order. |
if every element of $G$ is of finite order. |
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| All finite groups are periodic. |
All finite groups are periodic. |
| More generally, all locally finite groups are periodic. |
More generally, all locally finite groups are periodic. |
| Examples of periodic groups that are not locally finite include Tarski groups, |
Examples of periodic groups that are not locally finite include Tarski groups and infinite Burnside groups. |
| and Burnside groups $B(m,n)$ of odd exponent $n\ge665$ on $m>1$ generators. |
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| Some easy results on periodic groups: |
Some easy results on periodic groups: |
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| \begin{theorem} |
\begin{theorem} |
| \item Every \PMlinkname{subgroup}{Subgroup} of a periodic group is periodic. |
\item Every \PMlinkname{subgroup}{Subgroup} of a periodic group is periodic. |
| \end{theorem} |
\end{theorem} |
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| \begin{theorem} |
\begin{theorem} |
| \item Every \PMlinkname{quotient}{QuotientGroup} of a periodic group is periodic. |
\item Every \PMlinkname{quotient}{QuotientGroup} of a periodic group is periodic. |
| \end{theorem} |
\end{theorem} |
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| \begin{theorem} |
\begin{theorem} |
| \item Every \PMlinkname{extension}{GroupExtension} of a periodic group by a periodic group is periodic. |
\item Every \PMlinkname{extension}{GroupExtension} of a periodic group by a periodic group is periodic. |
| \end{theorem} |
\end{theorem} |
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| \begin{theorem} |
\begin{theorem} |
| \item Every restricted direct product of periodic groups is periodic. |
\item Every restricted direct product of periodic groups is periodic. |
| \end{theorem} |
\end{theorem} |
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| 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. |
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. |
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| Some further results on periodic groups: |
Some further results on periodic groups: |
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| \begin{theorem} |
\begin{theorem} |
| Every solvable periodic group is locally finite. |
Every solvable periodic group is locally finite. |
| \end{theorem} |
\end{theorem} |
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| \begin{theorem} |
\begin{theorem} |
| Every periodic abelian group is the direct sum of its maximal \PMlinkname{$p$-groups}{PGroup4} over all primes $p$. |
Every periodic abelian group is the direct sum of its maximal \PMlinkname{$p$-groups}{PGroup4} over all primes $p$. |
| \end{theorem} |
\end{theorem} |
