nLab torsion subgroup

Definition

Torsion and torsion-free classes of objects in an abelian category were introduced axiomatically as a torsion theory (or torsion pair) in (Dickson 1963).

Beware that there are other, completely independent, concepts referred to as torsion. See there for more.

In groups

The torsion subgroup of an abelian group $G \in$ Grp is the subgroup of all those elements $g \,\in\, G$, which have finite order, i.e. those for which some power is the neutral element

$$g \in G \;\text{is torsion} \;\;\;\;\;\;\Leftrightarrow\;\;\;\;\;\; \underset{n \in \mathbb{N}}{\exists} \;\;\; g^n \;\coloneqq\; \underset{ n\; factors }{ \underbrace{ g \cdot g \cdots g } } \,=\, \mathrm{e} \,.$$

A group is

• pure torsion if it coincides with its torsion subgroup,

• torsion-free if its torsion subgroup is trivial.

Notice that for abelian groups $A \in$ AbGrp the group operation is often written additively, namely as “$+$” and the neutral element is written as zero, whence this reads:

$$a \in A \;\text{is torsion} \;\;\;\;\;\Leftrightarrow\;\;\;\;\; \underset{n \in \mathbb{N}}{\exists} \;\;\; n \cdot a \;\coloneqq\; \underset{ n\; summands }{ \underbrace{ a + a \cdots + a } } \,=\, 0 \,.$$

In monoids

The situation with monoids is very similar to the situation with groups.

The torsion subgroup of a commutative monoid $M \in$ Mon is the submonoid of all those elements $m \,\in\, M$ for which some power is the neutral element

$$m \in M \;\text{is torsion} \;\;\;\;\;\;\Leftrightarrow\;\;\;\;\;\; \underset{n \in \mathbb{N}}{\exists} \;\;\; m^n \;\coloneqq\; \underset{ n\; factors }{ \underbrace{ m \cdot m \cdots m } } \,=\, \mathrm{e} \,.$$

Every such submonoid is a group, which is why the set of all such elements is called a torsion subgroup.

A monoid is

• pure torsion if it coincides with its torsion subgroup (and is thus the same as a pure torsion group),

• torsion-free if its torsion subgroup is trivial.

Notice that for commutative monoids $C \in$ CMon, where the monoid operation is traditionally written as addition $+$ and the neutral element is written as zero, this reads:

$$c \in C \;\text{is torsion} \;\;\;\;\;\Leftrightarrow\;\;\;\;\; \underset{n \in \mathbb{N}}{\exists} \;\;\; n \cdot c \;\coloneqq\; \underset{ n\; summands }{ \underbrace{ c + c \cdots + c } } \,=\, 0 \,.$$

In modules

Given a ring $R$, an element $m$ in an $R$-module $M$ is a torsion element if there is a nonzero element $r$ in $R$ such that $r m=0$. In constructive mathematics, given a ring $R$ with a tight apartness relation $\#$, an element $m$ in an $R$-module $M$ is a torsion element if there is a element $r$ in $R$ such that $r \# 0$ and $r m=0$.

A torsion module is a module whose elements are all torsion. A torsion-free module is a module whose elements are not torsion, other than $0$.

Properties

Relation to the $Tor$-functor

Proposition

For $A$ an abelian group, its torsion subgroup is isomorphic to the value of the degree-1 Tor functor $Tor^\mathbb{Z}_1(\mathbb{Q}/\mathbb{Z}, A)$.

See at Tor - relation to torsion subgroups for more.

Relation to flatness

Proposition

An abelian group is torsion-free precisely if regarded as a $\mathbb{Z}$-module it is a flat module.

This is a special case of a more general result for modules over a principal ideal domain. See also flat module - Examples for more. However, it is unclear whether this result holds in constructive mathematics, since the $\mathbb{Z}$ is not a principal ideal domain unless excluded middle holds.

Examples and applications

Example

(finite groups are pure torsion)
Every finite group is pure torsion, hence is its own maximal torsion subgroup.

Example

(tensoring with the rational numbers removes torsion subgroups)
Given an abelian group $A$ which is pure torsion (e.g. a finite abelian group, by Ex. ), its tensor product with the additive group of rational numbers is the trivial abelian group:

$$A \,\text{torsion} \;\;\; \Rightarrow \;\;\; A \otimes_{\mathbb{Z}} \mathbb{Q} \;\simeq\; 0 \;\;\; \in \; AbGrp \,.$$

Because, for $a \in A$ with $\underset{ n\;summands }{\underbrace{a + a + \cdots + a}} = 0$ and for $p,q \in \mathbb{Z} \subset \mathbb{Q}$ with $q \neq 0$ we have, by the definition of tensor product of abelian groups:

$$\begin{aligned} a \otimes \frac{p}{q} & \;=\; a \otimes \underset{n\; summands}{ \underbrace{ \Big( \frac{p}{n \cdot q} + \frac{p}{n \cdot q} + \cdot + \frac{p}{n \cdot q} \Big) }} \\ & \;=\; \underset{n\;summands}{ \underbrace{ \Big(a \otimes \frac{p}{n \cdot q}\Big) + \cdots + \Big(a \otimes \frac{p}{n \cdot q}\Big) }} \\ & \;=\; \underset{ n \; summands }{ \underbrace{ ( a + \cdots + a ) }} \otimes \frac{p}{n \cdot q} \\ & \;=\; 0 \otimes \frac{p}{n \cdot q} \\ & \;=\; 0 \,. \end{aligned}$$

In rational homotopy theory one considers the higher homotopy groups $\pi_n(X)$ of topological spaces $X$ tensored over $\mathbb{Q}$: the resulting groups $\pi_n(X) \otimes_{\mathbb{Z}} \mathbb{Q}$ are then necessarily torsion-free – in this sense rational homotopy theory studies spaces “up to torsion”.

Example

(torsion homotopy groups of spheres)
By the Serre finiteness theorem, the homotopy groups of spheres are finite groups, and hence pure torsion by Ex. , except in the degree of the dimension of the sphere and, for even-dimensional spheres, twice its dimension minus one:

$$\pi_k\big( S^n \big) \;\; \simeq \; \left\{ \begin{array}{ll} \mathbb{Z} & k = n \\ \mathbb{Z} \oplus \text{torsion} & k = 2n-1 \;\text{and}\; n\;\text{even} \\ \text{torsion} & \text{otherwise} \end{array} \right.$$

Example

(torsion in elliptic curves)
The torsion elements of an elliptic curve as a group are important in number theory and arithmetic geometry. See torsion points of an elliptic curve.

Related concepts

• torsion-free module

• p-torsion

• torsion approximation

• Pontryagin duality for torsion abelian groups

• Bockstein homomorphism

References

On axiomatization of torsion theory in abelian categories:

• Spencer Ernst Dickson, Torsion theories for abelian categories, Thesis, New Mexico State University (1963) (ProQuest)

• Spencer Ernst Dickson, A Torsion Theory for Abelian Categories, Transactions of the American Mathematical Society 121 1 (1966) 223-235 (doi:10.2307/1994341, jstor:1994341)