The torsionPlanetmathPlanetmath of a group G is the set

Tor(G)={gG:gn=e for some n}.

A group is said to be torsion-free if Tor(G)={e}, i.e. the torsion consists only of the identity elementMathworldPlanetmath.

If G is abelianMathworldPlanetmath (or, more generally, locally nilpotentPlanetmathPlanetmath) then Tor(G) is a subgroupMathworldPlanetmathPlanetmath (the torsion subgroup) of G. Whenever Tor(G) is a subgroup of G, then it is fully invariant and G/Tor(G) is torsion-free.

Example 1 (Torsion of a finite group)

For any finite groupMathworldPlanetmath G, Tor(G)=G.

Example 2 (Torsion of the circle group)

The torsion of the circle group R/Z is Tor(R/Z)=Q/Z.

Title torsion
Canonical name Torsion
Date of creation 2013-03-22 13:21:38
Last modified on 2013-03-22 13:21:38
Owner mhale (572)
Last modified by mhale (572)
Numerical id 8
Author mhale (572)
Entry type Definition
Classification msc 20K10
Synonym group torsion
Related topic PeriodicGroup
Defines torsion-free
Defines torsion group
Defines torsion subgroup
Defines torsion free