A Möbius map sending $\widehat C_{m,\alpha}$ into $\widehat C_{m,\beta}$ is $z\mapsto cz$ or $z\mapsto c/z$ with $c\ne0$

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A Möbius map sending $\widehat C_{m,\alpha}$ into $\widehat C_{m,\beta}$ is $z\mapsto cz$ or $z\mapsto c/z$ with $c\ne0$ · Prove2Me