torsion

The torsion of a group $G$ is the set

$$Tor(G)=\{g\in G:g^n=e\text{ for some }n\in \mathbb{N}\}.$$

A group is said to be 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 torsion subgroup) of $G$.Whenever $Tor(G)$ is a subgroup of $G$, then it is fully invariant and $G/Tor(G)$ is torsion-free.

Example 1 (Torsion of a finite group)

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

Example 2 (Torsion of the circle group)

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