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torsion (Definition)

The torsion of a group $ G$ is the set

$\displaystyle \mathop{\mathrm{Tor}}\nolimits (G) = \{ g \in G : g^n = e$   $\displaystyle \mbox{ for some $n \in \mathbb{N}$}$$\displaystyle \}. $
A group is said to be torsion-free if $ \mathop{\mathrm{Tor}}\nolimits (G) = \{e\}$, i.e. the torsion consists only of the identity element.

If $ G$ is abelian (or, more generally, locally nilpotent) then $ \mathop{\mathrm{Tor}}\nolimits (G)$ is a subgroup (the torsion subgroup) of $ G$. Whenever $ \mathop{\mathrm{Tor}}\nolimits (G)$ is a subgroup of $ G$, then it is fully invariant and $ G/\mathop{\mathrm{Tor}}\nolimits (G)$ is torsion-free.

Example 1 (Torsion of a finite group)
For any finite group $ G$, $ \mathop{\mathrm{Tor}}\nolimits (G) = G$.
Example 2 (Torsion of the circle group)
The torsion of the circle group $ \mathbb{R}/\mathbb{Z}$ is $ \mathop{\mathrm{Tor}}\nolimits (\mathbb{R}/\mathbb{Z}) = \mathbb{Q}/\mathbb{Z}$.



"torsion" is owned by mhale. [ full author list (2) ]
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See Also: periodic group

Other names:  group torsion
Also defines:  torsion-free, torsion group, torsion subgroup, torsion free
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Cross-references: circle, finite group, fully invariant, subgroup, locally nilpotent, abelian, identity element, group
There are 26 references to this entry.

This is version 5 of torsion, born on 2003-01-07, modified 2006-02-16.
Object id is 3885, canonical name is Torsion3.
Accessed 9932 times total.

Classification:
AMS MSC: 20K10 (Group theory and generalizations :: Abelian groups :: Torsion groups, primary groups and generalized primary groups)
Pending Errata and Addenda
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