nLab torsion subgroup

Contents

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 GG \in Grp is the subgroup of all those elements gGg \,\in\, G, which have finite order, i.e. those for which some power is the neutral element

gGis torsionng ngggnfactors=e. 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 AA \in AbGrp, where the group operation is traditionally written as addition ++ and the neutral element is written as zero, this reads:

aAis torsionnnaa+a+ansummands=0. 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 MM \in Mon is the submonoid of all those elements mMm \,\in\, M for which some power is the neutral element

mMis torsionnm nmmmnfactors=e. 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 CC \in CMon, where the monoid operation is traditionally written as addition ++ and the neutral element is written as zero, this reads:

cCis torsionnncc+c+cnsummands=0. 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 RR, an element mm in an RR-module MM is a torsion element if there is a nonzero element rr in RR such that rm=0r m=0. In constructive mathematics, given a ring RR with a tight apartness relation #\#, an element mm in an RR-module MM is a torsion element if there is a element rr in RR such that r#0r \# 0 and rm=0r 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 00.

Properties

Relation to the TorTor-functor

Proposition

For AA an abelian group, its torsion subgroup is isomorphic to the value of the degree-1 Tor functor Tor 1 (/,A)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 AA 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:

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

Because, for aAa \in A with a+a++ansummands=0\underset{ n\;summands }{\underbrace{a + a + \cdots + a}} = 0 and for p,qp,q \in \mathbb{Z} \subset \mathbb{Q} with q0q \neq 0 we have, by the definition of tensor product of abelian groups:

apq =a(pnq+pnq++pnq)nsummands =(apnq)++(apnq)nsummands =(a++a)nsummandspnq =0pnq =0. \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 π n(X)\pi_n(X) of topological spaces XX tensored over \mathbb{Q} : the resulting groups π n(X) \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:

π k(S n){ k=n torsion k=2n1andneven torsion otherwise \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.

References

On axiomatization of torsion theory in abelian categories:

Last revised on May 19, 2022 at 16:12:40. See the history of this page for a list of all contributions to it.