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Math Help - Smallest Group

  1. #1
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    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?
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  2. #2
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    Quote Originally Posted by bram kierkels View Post
    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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