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 $n\in ℕ$.

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 $rm=0$. 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 axomatically as a torsion theory (or torsion pair) in (Dickson).

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

## Properties

### Relation to the $\mathrm{Tor}$-functor

###### Proposition

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

See at Tor - relation to torsion subgroups for more.

### Relation to flatness

###### Proposition

An abelian group is torsion-free precisely if regarded as a $ℤ$-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 ${\pi }_{n}\left(X\right)$ of topological spaces $X$ tensored over $ℚ$: the resulting groups ${\pi }_{n}\left(X\right){\otimes }_{ℤ}ℚ$ 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 November 26, 2012 21:38:16 by Urs Schreiber (82.169.65.155)