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Thread: 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 $\displaystyle Z(D_n) $is trivial if n is odd and is {1, $\displaystyle 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 $\displaystyle b^ja^i $ is in $\displaystyle Z(D_n)$, then $\displaystyle g(b^ja^i)g^{-1} = b^ja^i $ for all g.
    if g=a then $\displaystyle a(b^ja^i)a^{-1} = b^ja^i $and if g=b $\displaystyle b(b^ja^i)b^{-1} = b^ja^i $
    but from here I'm quite stuck.

    Thanks!!
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  2. #2
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    I assume $\displaystyle a$ denotes a rotation and $\displaystyle b$ denotes a flip

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

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

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

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