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Math Help - dihedral group presentation proof

  1. #1
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    dihedral group presentation proof

    I'm given the following:

    D_{2n}=\langle r,s|r^n=s^2=e,rs=sr^{-1}\rangle

    This is a group presentation whose rigorous definition has something to do with quotient groups of free groups, or some such. However, I am only given the following non-rigorous definition:

    D_{2n} is the group generated by the objects r and s, which are known to satisfy the following relations: r^n=s^2=e and rs=sr^{-1} (where e is the identity).

    I'm then asked to find the order of the cyclic subgroup generated by r. In other words, what is the order of r?

    I just don't know how to make this work. Obviously the answer is n, but I don't know how to prove that from the available information. In fact, I suspect I may have somehow missed something. You can download a pdf of the textbook chapter here (the entire book can be had here, but it's in an unusual format called *.djvu). The exercise in question is #8 from that section (1.2).

    Thanks in advance!
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  2. #2
    MHF Contributor

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    if you look at the page 25, there are 6 facts about D_{2n}. the exercise want you to prove the first one, i.e. 1, r, \cdots , r^{n-1} are distinct and r^n = 1, which means the order of r is n.

    to prove this, you can only use the geometric description of r, i.e. a rotaion of a n-gon, which is fully explained at the beginning of the section.
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