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Math Help - center of the dihedral group

  1. #1
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    center of the dihedral group

    Hi,
    my question is, "for n = 3 or greater, show  Z(D_n)  is trivial if n is odd and is {1, a^{n/2} } if n is even."

    Any help would be appreciated; we haven't learned anything advanced in class yet, just about cosets and lagrange's thm.

    My initial attempt was to consider that if  b^ja^i is in  Z(D_n), then g(b^ja^i)g^{-1} = b^ja^i   for all g.
    if g=a then a(b^ja^i)a^{-1} = b^ja^i   and if g=b b(b^ja^i)b^{-1} = b^ja^i
    but from here I'm quite stuck.

    Thanks!!
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  2. #2
    Member Haven's Avatar
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    I assume  a denotes a rotation and b denotes a flip

    note that (a^ib^j)^{-1} = b^{-j}a^{-i} = b^{j}a^{-i}
    \Rightarrow a^ib^j = b^ja^{-i}

    So, we want to find all values of i and j such that: (a^ib^j)g = g(a^ib^j)

    if g= a^s
    then (a^ib^j)a^s = a^{i-s}b^j = a^{s+i}b^j
    \Rightarrow j \in \{ 0,1 \} and s+i \equiv i-s \bmod{n} \Rightarrow 2s \equiv 0 \pmod{n}
    Which means the s=\frac{n}{2}, which is only an integer when n is even.

    Now you can do the case if g= a^sb^t, You should get that j=0.
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