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Elements $g,h\in G$ are conjugate whenfor some $x\in G$. Thenfor every integer $n$, so $h^n=e$ exactly when $g^n=e$. They therefore have the same order.

The Möbius group consists of the Möbius transformationsof the Riemann sphere, under composition. If $h=xg{x}^{-1}$, thenThus $p$ is fixed by $g$ exactly when $x(p)$ is fixed by $h$. Conjugation gives a bijection between their fixed-point sets, so conjugate elements have the same number of fixed points.

A nonidentity Möbius transformation has at most two fixed points because its fixed-point equation is quadratic on the Riemann sphere. If it has only one repeated fixed point, conjugate that point to infinity; the transformation becomes a nontrivial translation $z\mapsto z+c$, which has infinite order. Consequently every nontrivial finite-order element has two distinct fixed points:This is fixed points of a finite-order Möbius transformation.