torsionTorsion32003-01-07 09:34:332006-02-16 04:36:43Definitiontorsion-freetorsion grouptorsion subgrouptorsion free\usepackage{amssymb}
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\newcommand*{\defn}[1]{\textbf{#1}}The \defn{torsion} of a group $G$ is the set
\[
\Tor(G) = \{ g \in G : g^n = e \mbox{ for some $n \in \Nset$}\}.
\]
A group is said to be \defn{torsion-free} if $\Tor(G) = \{e\}$,
i.e.\ the torsion consists only of the identity element.
If $G$ is abelian (or, more generally, locally nilpotent) then $\Tor(G)$ is a subgroup (the \defn{torsion subgroup}) of $G$.
Whenever $\Tor(G)$ is a subgroup of $G$, then it is fully invariant and $G/\Tor(G)$ is torsion-free.
\begin{example}[Torsion of a finite group]
For any finite group $G$, $\Tor(G) = G$.
\end{example}
\begin{example}[Torsion of the circle group]
The torsion of the circle group $\Rset/\Zset$ is $\Tor(\Rset/\Zset) = \Qset/\Zset$.
\end{example}