= Subgroup generated by complex scalings and reciprocal inversion
{title2=$z\mapsto\lambda z\quad\text{or}\quad z\mapsto\lambda/z$}
The <Möbius transformations> generated by $D_\lambda(z)=\lambda z$ with $\lambda\ne0$ and $I(z)=1/z$ have exactly the two displayed forms. The relations $I^2=1$ and $ID_\lambda=D_{\lambda^{-1}}I$ reduce every word to $D_\lambda I^\epsilon$, $\epsilon\in\{0,1\}$. This is a <semidirect product> of $\mathbb C^\times$ with a cyclic group of order two acting by multiplicative inversion. It preserves the unordered pair $\{0,\infty\}$, and only its scaling elements fix zero.
Back to article page
