Torsion (algebra) - Examples

Examples

1. Let M be a free module over any ring R. Then it follows immediately from the definitions that M is torsion-free (if the ring R is not a domain then torsion is considered with respect to the set S of non-zero divisors of R). In particular, any free abelian group is torsion-free and any vector space over a field K is torsion-free when viewed as the module over K.
2. By contrast with Example 1, any finite group (abelian or not) is periodic and finitely generated. Burnside's problem asks whether, conversely, any finitely generated periodic group must be finite. (The answer is "no" in general, even if the period is fixed.)
3. In the modular group, Γ obtained from the group SL(2,Z) of two by two integer matrices with unit determinant by factoring out its center, any nontrivial torsion element either has order two and is conjugate to the element S or has order three and is conjugate to the element ST. In this case, torsion elements do not form a subgroup, for example, S · ST = T, which has infinite order.
4. The abelian group Q/Z, consisting of the rational numbers (mod 1), is periodic, i.e. every element has finite order. Analogously, the module K(t)/K over the ring R = K of polynomials in one variable is pure torsion. Both these examples can be generalized as follows: if R is a commutative domain and Q is its field of fractions, then Q/R is a torsion R-module.
5. The torsion subgroup of (R/Z,+) is (Q/Z,+) while the groups (R,+),(Z,+) are torsion-free. The quotient of a torsion-free abelian group by a subgroup is torsion-free exactly when the subgroup is a pure subgroup.
6. Consider a linear operator L acting on a finite-dimensional vector space V. If we view V as an F-module in the natural way, then (as a result of many things, either simply by finite-dimensionality or as a consequence of the Cayley–Hamilton theorem), V is a torsion F-module.