= Möbius group
{c}
{title2=$\mathcal M\cong PSL_2(\mathbb C)$}
The Möbius group is the <group> of all <Möbius transformations> of the <Riemann sphere>, under <function composition>. An invertible $2\times2$ complex matrix represents $z\mapsto(az+b)/(cz+d)$, and nonzero scalar multiples represent the same transformation. Rescaling a representative to determinant one gives a surjective <group homomorphism> $SL_2(\mathbb C)\to\mathcal M$ with <kernel of a group homomorphism> $\{I,-I\}$. Thus it is also the <projective linear group> $PGL_2(\mathbb C)$ and the quotient $PSL_2(\mathbb C)=SL_2(\mathbb C)/\{\pm I\}$.
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