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Math Help - infinite dihedral group

  1. #1
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    infinite dihedral group

    "The infinite dihedral group
    Dis generated (as a subgroup of the group SR of
    bijections :
    R ! R), by the translation t(x) = x + 1 and the reflection s(x) = x of
    the real line. Work out its elements, and find the orbit and the stabilizer of each of

    the points 1, 1/2, 1/3"
    I have to deal with that problem but i have no idea that how i am going to use that points. I once read that ts(x)= -x+1 has reflection in the point 1/2 but i dont know how.
    Anyone to explain?
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  2. #2
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    Quote Originally Posted by dreamon View Post

    "The infinite dihedral group D∞ is generated (as a subgroup of the group SR of bijections : R ! R), by the translation t(x) = x + 1 and the reflection s(x) = −x of the real line. Work out its elements, and find the orbit and the stabilizer of each of the points 1, 1/2, 1/3".

    I have to deal with that problem but i have no idea that how i am going to use that points. I once read that ts(x)= -x+1 has reflection in the point 1/2 but i dont know how.
    Anyone to explain?
    first see that tst=s and thus st=t^{-1}s, which implies that st^i=t^{-i}s, for all i \in \mathbb{Z}. therefore D_{\infty}=\{t^is^j: \ i \in \mathbb{Z}, \ j \in \{0,1\} \}. for the second part of your question, to find the orbit of a \in \mathbb{R},

    we need to find t^is^j(a), for any i \in \mathbb{Z}, \ j \in \{0,1\}. we have: s^0(a)=a, \ s(a)=-a. so we can write s^j(a)=(-1)^ja. thus t^is^j(a)=t^i((-1)^ja)=(-1)^ja + i. therefore the orbit of a is the set

    \{(-1)^ja + i : \ i \in \mathbb{Z}, \ j \in \{0,1 \} \}. in order to find he stabilizer of a we need to find all i \in \mathbb{Z}, \ j \in \{0,1 \} such that (-1)^ja+i=a. that will depend on a. for example, if a=1, then we need to

    solve (-1)^j + i = 1. if j=0, then we have i = 0 and if j=1, then i=2. so the stabilizer of a=1 is \{1, t^2s \}.

    for a=1/2 we need to solve (-1)^j/2 + i= 1/2, which can be written as  (-1)^j + 2i = 1. now if j=0, then i=0 and if j=1, we'll get i=1. so the stabilizer of a=1/2 is \{1,ts \}.
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