Smallest Group

**bram kierkels** · September 14th 2010, 01:19 AM

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?

**tonio** · September 14th 2010, 06:59 AM

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

Tonio

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