# Smallest Group

• Sep 14th 2010, 02:19 AM
bram kierkels
Smallest Group
I have two maps $m,t:\mathbb{C}\rightarrow\mathbb{C},$:

$m(z)=iz \ \ \ \ \ \ \ \ \ t(z)=1+z$

I want to find the smallest group of transformations of $\mathbb{C}$ that contains both m and t.

Any hints?
• Sep 14th 2010, 07:59 AM
tonio
Quote:

Originally Posted by bram kierkels
I have two maps $m,t:\mathbb{C}\rightarrow\mathbb{C},$:

$m(z)=iz \ \ \ \ \ \ \ \ \ t(z)=1+z$

I want to find the smallest group of transformations of $\mathbb{C}$ that contains both m and t.
Any hints?

$m^2(z)=i(iz)=-z\,,\,\,m^3(z)=i(-z)=-iz\,,\,\,m^4(z)=i(-iz)=z\Longrightarrow m^4=Id.$ , whereas $t^2(z)=2+z\,,\ldots,t^k(z)=k+z\,,\,\forall\,k\in\m athbb{Z}$

Thus, the group $$ is infinite (since the order of t is), and since m, t don't commute...

Tonio