nLab
torsion subgroup

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Definition

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

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

Given a ring RR, an element mm in an RR-module MM is torsion element if there is a nonzero element rr in RR such that 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.

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 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.

Examples and applications

References

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

Revised on December 4, 2013 23:43:09 by David Corfield (91.125.226.129)