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torsion
The torsion of a group $G$ is the set$$ \Tor(G) = \{ g \in G : g^n = e \mbox{ for some $n \in \Nset$}\}.$$ A group is said to be torsion-free if $\Tor(G) = \{e\}$ , i.e. the torsion consists only of the identity element.
If $G$ is abelian (or, more generally, locally nilpotent) then $\Tor(G)$ is a subgroup (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 group $G$ , $\Tor(G) = G$ .
For any finite group $G$ , $\Tor(G) = G$ .
Example 2 (Torsion of the circle group)
The torsion of the circle group $\Rset/\Zset$ is $\Tor(\Rset/\Zset) = \Qset/\Zset$ .
The torsion of the circle group $\Rset/\Zset$ is $\Tor(\Rset/\Zset) = \Qset/\Zset$ .
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