Contents

group theory

# Contents

## 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 a 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, where the group operation is traditionally written as addition $+$ and the neutral element is written as zero, 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 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.

On axiomatization of torsion theory in abelian categories: