torsion subgroup

- group, ∞-group
- group object, group object in an (∞,1)-category
- abelian group, spectrum
- group action, ∞-action
- representation, ∞-representation
- progroup
- homogeneous space

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.

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.

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.

- 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”.

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

Revised on February 18, 2018 09:30:26
by Urs Schreiber
(195.229.110.1)