= Torsion-freeness from one nonidentity conjugacy class
If an infinite <group> has a single nonidentity <conjugacy class>, it is a <torsion-free group>. Otherwise every nonidentity element has the same finite order $n$. Powers show $n$ is prime. For $n\geq3$, choose $xgx^{-1}=g^2$; then $x^n=1$ gives $g=g^{2^n}$, impossible by <Fermat's little theorem>. For $n=2$, the group has exponent two and is abelian, so every conjugacy class is a singleton and the hypothesis would force the group to have only two elements. Infinitude excludes this final case.
Back to article page
