I have two maps $\displaystyle m,t:\mathbb{C}\rightarrow\mathbb{C},$:
$\displaystyle m(z)=iz \ \ \ \ \ \ \ \ \ t(z)=1+z$
I want to find the smallest group of transformations of $\displaystyle \mathbb{C}$ that contains both m and t.
Any hints?
I have two maps $\displaystyle m,t:\mathbb{C}\rightarrow\mathbb{C},$:
$\displaystyle m(z)=iz \ \ \ \ \ \ \ \ \ t(z)=1+z$
I want to find the smallest group of transformations of $\displaystyle \mathbb{C}$ that contains both m and t.
Any hints?
$\displaystyle m^2(z)=i(iz)=-z\,,\,\,m^3(z)=i(-z)=-iz\,,\,\,m^4(z)=i(-iz)=z\Longrightarrow m^4=Id.$ , whereas $\displaystyle t^2(z)=2+z\,,\ldots,t^k(z)=k+z\,,\,\forall\,k\in\m athbb{Z}$
Thus, the group $\displaystyle <m,t>$ is infinite (since the order of t is), and since m, t don't commute...
Tonio