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Math Help - Dihedral Groups

  1. #1
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    Dihedral Groups

    What is the center of D_n?

    It seems to me that when n is odd the center is e, and when n is even there are some rotations included.

    Any help is great.
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  2. #2
    MHF Contributor kalagota's Avatar
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    you're right. if n is odd, the center is e.
    now, if n is even, the center is \{e, a^{n/2}\}. this should be easy to prove.
    Last edited by kalagota; April 7th 2009 at 07:06 AM.
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  3. #3
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    Quote Originally Posted by kalagota View Post

    now, if n is even, the center is \{e, a^{n/2}b\}. this should be easy to prove.
    i think you meant \{e, a^{n/2} \} and the proof is not hard but requires some effort!

    Edit: note that D_2 is abelian and thus it's equal to its center. so if n > 2 is even, then the center of D_n is \{e, a^{n/2} \}.
    Last edited by NonCommAlg; April 6th 2009 at 10:44 AM.
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    Thanks for taking the time to answer my question. I just have a question on the notation that you used, what is a^{n/2}? does this represent a "flip" or a rotation. Also how would I begin to prove that this is abelian to every element in D_n where n is even? Should i use a multiplication table?
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  5. #5
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    Quote Originally Posted by Order 2 View Post
    Thanks for taking the time to answer my question. I just have a question on the notation that you used, what is a^{n/2}? does this represent a "flip" or a rotation. Also how would I begin to prove that this is abelian to every element in D_n where n is even? Should i use a multiplication table?
    recall that D_n=<a,b: \ a^n=b^2=1, \ ba=a^{-1}b>=\{1,a, \cdots , a^{n-1}, b, ab, a^2b , \cdots , a^{n-1}b \}. since a,b generate D_n, an element x=a^ib^j, \ 0 \leq i \leq n-1, \ 0 \leq j \leq 1, of D_n is in the center of

    D_n if and only if ax=xa and bx=xb. this will simplify your job a lot!
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  6. #6
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by NonCommAlg View Post
    i think you meant \{e, a^{n/2} \} and the proof is not hard but requires some effort!
    yeah, that's what i meant. tnx! i edited it already!
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