Torsion element

A torsion element has finite order.

A group is a torsion group when every element is a torsion element, meaning that each element has finite order. The orders need not have a common finite bound. A torsion group has no nontrivial torsion-free group as a subgroup; in particular it cannot contain a nonabelian free group. Infinite finitely generated groups of this kind can be constructed using a torsion group construction by p-power relators.

A group is locally finite when every subgroup that is a finitely generated group is finite. It is a torsion group, since the cyclic subgroup generated by one element must be finite. A restricted direct sum of groups with finite factors is locally finite. A locally finite group can be infinite, while a locally finite finitely generated group is finite by definition.

The torsion subgroup of an abelian group consists of all elements of finite order. For a finitely generated abelian group, it is the finite direct summand complementary to the free abelian part.

For an abelian group $A$, its $p$-primary torsion subgroup is $A\{p\}=\{a:p^{r}a=0\text{ for some }r\ge 0\}$. It is zero exactly when $A[p]=0$, since a nonzero element of $p$-power order has a multiple of order $p$. Multiplication by an integer prime to $p$ is an automorphism on it.