group theory

# Contents

## Definition

The torsion subgroup of a group is the subgroup of all those elements $g$, which have finite order, i.e. those for which $g^n = e$ for some $n \in \mathbb{N}$.

A group is torsion-free if there is no such element apart from the neutral element $e$ itself, i.e. when the torsion subgroup is trivial.

Given a ring $R$, an element $m$ in an $R$-module $M$ is torsion element if there is a nonzero element $r$ in $R$ such that $r m=0$. 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 $0$.

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

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

## 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.

## Examples and applications

• In rational homotopy theory one considers the 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”.

## References

• S. E. Dickson, Torsion theories for abelian categories, Thesis, New Mexico State University (1963).

Revised on August 26, 2014 23:15:51 by Urs Schreiber (82.136.246.44)