# torsion

The of a group $G$ is the set

 $\mathop{\mathrm{Tor}}\nolimits(G)=\{g\in G:g^{n}=e\mbox{ for some n\in\mathbb% {N}}\}.$

A group is said to be torsion-free if $\mathop{\mathrm{Tor}}\nolimits(G)=\{e\}$, i.e. the torsion consists only of the identity element.

If $G$ is abelian (or, more generally, locally nilpotent) then $\mathop{\mathrm{Tor}}\nolimits(G)$ is a subgroup (the torsion subgroup) of $G$. Whenever $\mathop{\mathrm{Tor}}\nolimits(G)$ is a subgroup of $G$, then it is fully invariant and $G/\mathop{\mathrm{Tor}}\nolimits(G)$ is torsion-free.

###### Example 1 (Torsion of a finite group)

For any finite group $G$, $\mathop{\mathrm{Tor}}\nolimits(G)=G$.

###### Example 2 (Torsion of the circle group)

The torsion of the circle group $\mathbb{R}/\mathbb{Z}$ is $\mathop{\mathrm{Tor}}\nolimits(\mathbb{R}/\mathbb{Z})=\mathbb{Q}/\mathbb{Z}$.

 Title torsion Canonical name Torsion Date of creation 2013-03-22 13:21:38 Last modified on 2013-03-22 13:21:38 Owner mhale (572) Last modified by mhale (572) Numerical id 8 Author mhale (572) Entry type Definition Classification msc 20K10 Synonym group torsion Related topic PeriodicGroup Defines torsion-free Defines torsion group Defines torsion subgroup Defines torsion free